Gollum is a slimy, skinny creature living in the middle of an underground lake underneath the Misty Mountains. As Tolkien describes him, he is "dark as darkness, except for two big round pale eyes in his thin face" (5.12). He has spent so much time underground that his eyes have grown giant and protruding on account of all of his peering through murky tunnels. Gollum spends a lot of time alone, something we can observe pretty easily by the frequency with which he talks to himself. He also punctuates his whispering with noisy gulping swallows, hence the name, "Gollum."
Gollum is quite dangerous to Bilbo, since he's fond of eating other people and sometimes gets tired of fish. But he's obviously lonely and sad: he talks to himself using "we," so he must want company. He's not just a threat; he is also incredibly pitiful.
Gollum has one treasure, a golden ring he calls his "birthday-present" (5.100) because it came to him – you guessed it – on his birthday. This ring makes the wearer invisible. Gollum uses it to sneak around the goblin tunnels and occasionally catch young goblins to eat (ick!). How it came to him in the first place, we have to wait until The Lord of the Rings to find out. The narrator comments only, "who knows how Gollum came by that present, ages ago in the old days when such rings were still at large in the world?" (5.78). All we can say for now is that Gollum has dropped it in the goblin tunnels, and Bilbo just happens to pick it up again.
When Gollum first encounters Bilbo, he isn't too hungry for hobbit-flesh, but he's rather afraid of Bilbo's knife. So he tries to get on Bilbo's good side by challenging him to a friendly game of riddles (since "the riddle-game was sacred and of immense antiquity"). Bilbo agrees, because he wants Gollum to show him the way out of the goblin kingdom under the Misty Mountains. But the game makes Gollum both hungry and angry.
Gollum tries to find his ring so he can become invisible and kill Bilbo. Bilbo, of course, being the luckiest person in the world, just happens to put on Gollum's ring right as Gollum comes to attack him. So he becomes invisible and follows Gollum out of the goblin tunnels. As Bilbo takes his final look at Gollum, he almost kills the wretched creature. But Bilbo is filled with a "sudden understanding, a pity mixed with horror." He sees Gollum's "endless, unmarked days without light or hope of betterment" (5.119). And Bilbo lets Gollum live. In exchange for this kindness, Gollum screams, "Thief, thief, thief! Baggins! We hates it, we hates it, we hates it for ever!" (5.122) as Bilbo makes his escape.
Bilbo's decision to let Gollum live has a huge effect on the Middle-earth stories to come. Gollum and Gandalf both have much expanded roles in The Lord of the Rings. With this, it's really in that series that we learn more about the complexities of these characters. In fact, after publishing The Lord of the Rings, Tolkien revised Bilbo's encounter with Gollum in The Hobbit to bring it more in line with the developments in that work. So the 1937 Gollum is much less pathetic than the Gollum we find in the revised edition of 1966. To see an extended comparison of the 1937 and 1966 texts, check out Douglas A. Anderson's amazing volume, The Annotated Hobbit.