Smaug is a "most especially greedy, strong, and wicked" (1.123) dragon who comes flying out of the North to attack the Lonely Mountain a couple of generations before The Hobbit takes place. He's attracted to this particular dwarf kingdom because it's an especially wealthy one in the days of Thorin's grandfather, Thror. And we all know that dragons love to sit on top of piles of treasure. So Smaug flies in one day, eats pretty much all the dwarves inside their tunnels, and settles down on top of his stolen hoard of gold and silver. Smaug also destroys the nearby human town of Dale (where Bard's ancestor Girion was lord). So Smaug has a lot of blood on his claws – boo.
Bilbo first encounters Smaug when he creeps down the tiny side passage into the Lonely Mountain and steals a golden cup from Smaug's stack while the dragon is sleeping. Even though Smaug isn't doing anything with this treasure, he jealously guards every tiny bit of it, and he immediately realizes that the cup has been stolen. In retaliation, Smaug tears up the side of the mountain and blocks off the tiny side door (even though he can't see quite where it is). So, this Smaug creature likes to overreact. And his overreaction gets even worse after his second visit from Bilbo.
Bilbo creeps down the passage a second time to see if he can find some kind of weak point in Smaug's armor. This time, Smaug is awake. Luckily, Bilbo's ring of invisibility gives him some safety. And Smaug wants to find out where Bilbo is from, so he doesn't kill him outright. They fall into conversation. Smaug asks Bilbo's name. Bilbo doesn't want to give Smaug his name for fear of evil spells, but he also doesn't want to refuse Smaug's request for fear of angering the dragon (neither would we). So Bilbo calls himself a number of fancy names – "Barrel-rider" being the primary one. Bilbo tells Smaug that he isn't here for money; he and his friends have come for revenge. Smaug laughs at the very idea. He turns over to show off the magnificence of his scales. He thinks no weapon will be able to hurt him. But Smaug is unaware of an unscaled patch on his chest: a weak spot! Bilbo creeps off, and Smaug sends a wave of flame after him. Bilbo barely escapes unburned.
This meeting has pros and cons for both Smaug and Bilbo. On Smaug's side, he has figured out that Bilbo must have come from Lake-town in the company of some dwarves. So Smaug flies out to destroy Lake-town. Bilbo, for his part, has found out that Smaug has a weak point. But he has also accidentally endangers Lake-town. Luckily, a helpful thrush carries this news about the weak point to Bard, the heroic archer of Lake-town who then brings down Smaug. (Unfortunately, Smaug's giant body falls right on top of Lake-town and crushes it, but we suppose he can't be blamed for that one.)
In addition to the obvious dragon qualities of greed, murderousness, and cruelty, Tolkien's Smaug shows us one more, slightly unusual aspect of dragon character. According to Tolkien, dragons are incredibly persuasive. Just by talking to Bilbo, Smaug almost manages to convince him that the dwarves are secretly planning some kind of double-cross of their trusty burglar.
As with pretty much everything in Tolkien, this idea of dragons as silver-tongued has a long history in folklore. This whole conversation between Bilbo and Smaug seems to be based on The Song of Fáfnir (Fáfnismál) in the Poetic Edda, a collection of Old Norse songs (source: Tom Shippey, The Road to Middle-earth. How J.R.R. Tolkien Created a New Mythology. New York: HarperCollins, 1998, pg. 283). So if you desperately need more dragon-human contact, either check out The Song of Fáfnir or take a gander at Siegfried, the third of Wagner's massively long Ring cycle of operas.