If you've read The Lord of the Rings and were intrigued by its glimpses of Middle-earth's history and legends, you might very much enjoy delving deeper by reading The Children of Húrin, the most fully developed of those tales. Like most stories of that earlier age, it is a tragedy filled with loss and sorrow, but a good read nonetheless.
My goal in writing this "bridge" is to help you jump straight into that book even without reading The Silmarillion first to understand its background. Christopher Tolkien's introduction to The Children of Húrin does this, too, but it's fifteen pages long and very thorough: it really is a fantastic overview, but only if you've got the patience for that much rather dry historical summary. If you'd rather get straight to the story and pick up more details from context, this brief bridge is for you. 
The good news is that we can mostly set the stage for The Children of Húrin with a few reminders of passages where The Lord of the Rings already referred to Middle-earth's ancient history. Elrond himself makes a good start on this during the Council, just after he first describes the assembled hosts of the Last Alliance of Elves and Men:
Thereupon Elrond paused a while and sighed. 'I remember well the splendour of their banners,' he said. 'It recalled to me the glory of the Elder Days and the hosts of Beleriand, so many great princes and captains were assembled. And yet not so many, nor so fair, as when Thangorodrim was broken, and the Elves deemed that evil was ended for ever, and it was not so.'
'You remember?' said Frodo, speaking his thought aloud in his astonishment. 'But I thought,' he stammered as Elrond turned towards him, 'I thought that the fall of Gil-galad was a long age ago.'
'So it was indeed,' answered Elrond gravely. 'But my memory reaches back even to the Elder Days. Eärendil was my sire, who was born in Gondolin before its fall; and my mother was Elwing, daughter of Dior, son of Lúthien of Doriath. I have seen three ages in the West of the world, and many defeats, and many fruitless victories.'
The fall of Gil-Galad and the victory of the Last Alliance over Sauron took place at the end of the Second Age, a little over 3000 years before The Lord of the Rings. Elrond's memory of the even greater battle when the mountain towers of Thangorodrim were broken at the end of the First Age took place nearly 3500 years earlier. The Children of Húrin takes place about 100 years before that, not long before Elrond was born, in the land of Beleriand long since lost beneath the waves west of the Grey Havens. The famous hidden city of Gondolin where Elrond's father was born plays a role early in Húrin's story, and major parts of his children's story take place in and around Doriath where Lúthien Tinúviel once lived.
Elrond's reminiscence wasn't the first story of the Elder Days that the hobbits had heard. While they waited for nightfall in the shadow of Weathertop, Aragorn sang a part of the tale of Tinúviel: that story took place in the very year that Túrin son of Húrin was born. Aragorn's summary of the tale is a good glimpse of the state of the world when The Children of Húrin begins (but I've added some paragraph breaks: what a wall of text!):
It tells of the meeting of Beren son of Barahir and Lúthien Tinúviel. Beren was a mortal man, but Lúthien was the daughter of Thingol, a King of Elves upon Middle-earth when the world was young; and she was the fairest maiden that has ever been among all the children of this world. As the stars above the mists of the Northern lands was her loveliness, and in her face was a shining light. In those days the Great Enemy, of whom Sauron of Mordor was but a servant, dwelt in Angband in the North, and the Elves of the West coming back to Middle-earth made war upon him to regain the Silmarils which he had stolen; and the fathers of Men aided the Elves.
But the Enemy was victorious and Barahir was slain, and Beren escaping through great peril came over the Mountains of Terror into the hidden Kingdom of Thingol in the forest of Neldoreth. There he beheld Lúthien singing and dancing in a glade beside the enchanted river Esgalduin; and he named her Tinúviel, that is Nightingale in the language of old. Many sorrows befell them afterwards, and they were parted long. Tinúviel rescued Beren from the dungeons of Sauron, and together they passed through great dangers, and cast down even the Great Enemy from his throne, and took from his iron crown one of the three Silmarils, brightest of all jewels, to be the bride-price of Lúthien to Thingol her father.
Yet at the last Beren was slain by the Wolf that came from the gates of Angband, and he died in the arms of Tinúviel. But she chose mortality, and to die from the world, so that she might follow him; and it is sung that they met again beyond the Sundering Seas, and after a brief time walking alive once more in the green woods, together they passed, long ago, beyond the confines of this world.'
Many of the people and places in Aragorn's story reappear in the story of Húrin and his children; you can find them on the map below. Húrin's grandfather was the first Lord of the land of Dor-lómin, which was not far from the evil fortress of Angband in the north whose gates were guarded by the mountain towers of Thangorodrim. Their family were the leaders of a tribe of humans (or Edain) who served for years as vassals of the king of the High Elves (or Noldor), the Elves of the West who had returned to Middle-earth to recover their stolen Silmarils from Morgoth the Great Enemy. Húrin's wife Morwen is a relative of Beren's, and when their son Túrin needs a place of safety in the book he is fostered by King Thingol in Beren's honor. Thingol's people in the kingdom of Doriath were mostly Sindar: Elves somewhat related to the Wood-elves of The Hobbit. (The term Eldar refers to all of these various Elves together.)
That is probably the bare minimum background that you need to begin reading The Children of Húrin. It can be very helpful while reading to refer to the detailed map in the book: in my copy there's a fold-out map just before the back of the book (my map below only labels places from this "bridge"). There's a useful glossary of names in the back as well. If you're up for it, I've given a few additional details below (still much less than Christopher Tolkien's full introduction), but if you're willing to pick up most of the names and setting from context then you may be ready to plunge straight into the book. I hope that you enjoy it! And then if you someday read The Lord of the Rings again, Elrond's words of praise after Frodo offers to carry the Ring to Mt. Doom will carry much deeper resonance for you: "though all the mighty Elf-friends of old, Hador, and Húrin, and Túrin, and Beren himself were assembled together, your seat should be among them."
If you're up for slightly more detailed context, we can use the map above to inspire some discussion of people and history. (Again, you'll want to follow the original, fully detailed map when reading the actual book. Mine is just a sketch, and omits a lot.) The map shows part of Beleriand, which was the westernmost region of Middle-earth until it drowned beneath the sea, long before the time of The Lord of the Rings. (Compared to the LotR map, everything on this map is west of the mountains around the Grey Havens, sunk under the waves.)
As a starting point, Aragorn refers to "the Elves of the West coming back to Middle-earth": this passing comment introduces an important bit of history. Before that return, Beleriand was populated by Sindarin and Silvan Elves, not unlike the Wood-elves in The Hobbit. The forested land of Doriath in the center of the map was their greatest realm, ruled by King Thingol and his wife Melian, parents of Lúthien. Melian was not an Elf but rather one of the Maiar, a powerful spirit who had taken the form of a woman, and she used her magic to create what Christopher Tolkien described as "an unseen wall of shadow and bewilderment" around Doriath (it was called "the Girdle of Melian") that kept enemies and strangers from finding their way in.
The Elves of the West (more formally, the Noldor) who came back to Middle-earth were often called High Elves in The Lord of the Rings: Glorfindel was one of them, as were Gildor and his companions who met Frodo in the Shire. Because the Noldor had returned to Middle-earth to make war on Morgoth, the Great Enemy, they established realms of their own in the lands surrounding his fortress of Angband in the far north, as shown on the map. (The gate of that fortress opened at the base of the three mountain towers of Thangorodrim: these belong somewhere around the top edge of the official Children of Húrin map; I've included them in roughly the right spot on my mine.) Although there was some mistrust between the Noldor and the Sindar in Doriath, they recognized each other as a common people (the Eldar) and they were all foes of Morgoth.
The High King of the Noldor was Fingolfin; he and his son Fingon ruled over Hithlum, ringed by mountains in the northwest. (I'm very sorry that so many Elves' names start with "Fin-": a family tradition, I'm afraid.) Húrin and Túrin's story eventually began there, in the part of Hithlum called Dor-lómin. Fingolfin's second son Turgon founded the hidden city of Gondolin, built on a plateau encircled by the tall mountains north of Doriath where it was guarded by the Great Eagles: no outsider guessed where his kingdom lay. Fingolfin's nephew Finrod Felagund (brother of Galadriel) built a secret fortress of his own, Nargothrond, in the hills west of Doriath's southern border. The High Elves of the West had other realms as well, but they do not play as great a role in this tale.
Humans did not join the Elves in Beleriand until 300 years after the Noldor had returned from the West, and our story takes place almost 200 years after that. For generations, three human Houses (collectively called the Edain, or sometimes the Elf-friends) had lived as vassals of various Elven lords. Húrin's family ruled the House of Hador: they served High King Fingolfin as lords of Dor-lómin, the southern region of Hithlum in the northwest. The House of Bëor dwelt in the highlands of Dorthonion north of Doriath (separated from it by the impassible Mountains of Terror, infested with giant spiders akin to Shelob): Beren was of that people, as was Húrin's wife. And The House of Haleth had eventually settled in the forest of Brethil just northwest of Doriath: Húrin's mother was of that kin.
Those are the most important peoples and places you'll need to know before reading The Children of Húrin. Two points of recent history form a backdrop for the events of the story. First, nine years before Túrin son of Húrin's birth, the Great Enemy Morgoth launched a massive surprise assault against the Elves and Men who stood guard around his fortress. This was the Dagor Bragollach, the Battle of Sudden Flame, when rivers of fire burned the plain of Ard-galen to ash and Glaurung Father of Dragons first came forth in all his power at the head of an army of Orcs, who conquered everything north of Doriath and east of Hithlum (but not Gondolin). (When Aragorn said, "the Enemy was victorious", this is what he was talking about.) After the disaster, High King Fingolfin challenged Morgoth to hopeless single combat in fury and dispair and was slain, leaving Fingon as High King. Later, very near the time of Túrin's birth, Beren and Lúthien undertook their quest to reclaim a Silmaril from Morgoth's crown as described in Aragorn's story. During that quest, Finrod King of Nargothrond died to save Beren's life in the dungeons of Sauron, Morgoth's chief servant; Finrod's brother Orodreth then became lord of Nargothrond.
And with that, you should have more or less the background you need to begin reading The Children of Húrin. (You probably don't even need to remember all the details: just be ready to recognize them when they turn up.) I hope that you enjoy it!
 If you've got the patience for a little bit of dry background summary, the first few paragraphs of Appendix A of LotR give a nice quick overview of the history of the First Age. Start reading after the heading "I. The Númenórian Kings: (i) Númenor" where it begins, "Fëanor was the greatest of the Eldar in arts and lore," and read five paragraphs until a blank line (ending with "Of these things the full tale, and much else concerning Elves and Men, is told in The Silmarillion.").
And incidentally, even if you don't enjoy that sort of dry historical summary, I hope you've already read the lovely narrative story in Appendix A.I(v) a little later: "Here follows a part of the tale of Aragorn and Arwen".
 A number of people have tried to reconstruct how the Beleriand map fits with the Lord of the Rings map. It's a little ambiguous, since Tolkien never published a map with them together, but roughly speaking the whole Children of Húrin map would be under the sea west and north of the Grey Havens. The two reconstructions on this page are pretty similar and quite good. (An accurate map should show the Blue Mountains about the same distance from the Sea as from the Misty Mountains: a lot of combined maps out there make Beleriand much too wide. Ignore any map that looks like a pair of lungs: those are embarrassingly wrong.) The southern edge of the Children of Húrin map shows the mouths of the river Sirion as it reaches the sea. The adjacent coastline that runs east-west is more or less due west of what are later the southernmost coast of the Gulf of Lune, Sarn Ford (south of the Shire), and Eregion.