Partridge is your classic sidekick. In fact, he's so classic that the description of the "Bumbling Sidekick" that comes up in TV Tropes might have been made for him: he's an "annoying, incompetent sidekick for another character." His general uselessness not only makes Tom look good, but it also provides some comic relief for the novel as a whole.
Partridge starts out as a local schoolmaster on Squire Allworthy's estate. Even in these early days of the plot, the narrator is pretty up front about what kind of a guy he is: he is "one of the best-natured fellows in the world," and a "master of […] pleasantry and humour" (2.3.2). In other words, he's a jolly, fun-loving fellow (but not too bright). He's not so good at the school part of his job, being not all that learned. (In fact, the running joke of Partridge's bad Latin continues on throughout the novel.) But he's popular because he's always got a joke at hand.
Partridge's life gets ruined early on by his horribly jealous wife, who has all kinds of crazy suspicions about her husband and Jenny Jones. Their disagreements insert early slapstick humor into the book, as when Mrs. Partridge attacks Mr. Partridge in such a frenzy that her blouse falls open and she faints. Poor Mr. Partridge is clearly not going to be the hero of this tale, since he gets beaten up several times by women and since he can think of no defense against the accusation that he is Tom's out-of-wedlock father (which he isn't).
Like Jenny Jones, Partridge disappears from the novel early on, after Squire Allworthy's decision that he has to be guilty of fathering a bastard. But he comes back into the plot under the fake name of "Little Benjamin," as he works as Tom's barber/surgeon after Tom's injury by Ensign Northerton at the inn at Gloucester.
Partridge soon gives up the Little Benjamin thing to become Tom's assistant on his travels. After all, every decent hero needs a good-natured but dumb companion bumbling along with him to make him look good, right?
By the end of the novel, Partridge's faithful (though often fumbled) support of Tom pays off, when he gets to return to Somerset with a regular stipend. We're glad to know that Partridge might wind up married, now that Sophia is helping him sort out his romantic life and now that he has a steady pension from Tom. But—to Molly Seagrim? Does that seem random to anybody else? Isn't she at least twenty-five years younger than him? We get that Fielding is trying to tie off loose ends, but this one seems like kind of a stretch.
Partridge has at least three characteristics that mark him as a comic figure:
When Tom and Partridge get lost at night in Book 12, Chapter 11, Partridge believes that a witch has made them lose their way. No, really, he truly convinces himself that it's more likely that a witch has interfered with their travels than that they've just missed the road to Coventry in the darkness and bad weather.
And later on, when Tom finds out that Partridge has told Mrs. Miller of Tom's relationship to Squire Allworthy (against Tom's express wishes), Partridge claims that maybe that witch from Book 12 did it. Tom just laughs at Partridge's lame excuse, but the fact that Partridge thought that excuse had a chance of working proves his real faith in the supernatural.
Partridge's fear of ghosts, witches, and things that generally go bump in the night prove that he is an ignorant and cowardly guy. In Book 8, Chapter 1, the narrator comes out and says that any modern author who dares to include a real ghost in his novel risks a "horse-laugh" (8.1.3) from his audience.
Partridge cannot shut his mouth, even after Tom asks him to repeatedly. Thanks to Partridge's loose lips, Sophia thinks that Tom has been bad-mouthing her in every inn in southern England. But even worse, it's Partridge who tells Squire Allworthy straight to his face that Tom slept with Jenny Jones (his supposed mother). This is even after Partridge warns Mrs. Miller not to let on to Squire Allworthy that (again, supposedly) Tom has committed incest with his mother. It's like Partridge can't help himself: he just gets caught up in the moment, and he can't keep a secret to save his life.
This trait may just be a down side of Partridge's generally social and friendly nature. After all, the narrator points out that Partridge can't hold in a joke, "without the least respect of persons, time, or place" (8.4.4). Clearly, Partridge is just a guy without any judgment—he can't stop talking, be it gossip, jokes, or fearful ramblings about ghosts and goblins. Partridge's lack of self-control makes Tom's discipline problems look mild in comparison.
This trait isn't really as funny as Partridge's superstition and gossip, but Fielding mostly keeps it light. There are other characters in the book—Mr. Blifil and Lady Bellaston come to mind—whose selfishness has really dire consequences. Partridge's self-centeredness is mostly harmless and, therefore, humorous. His motivations are so transparent that, even if he's a bit immoral sometimes (since he's willing to lie and steal if Tom says it's okay), he never seems as dangerous as the deceptive Mr. Blifil.
So, for example, when Tom decides to go join the army of King George II, Partridge goes off on this long rant about how un-Christian it is to fight your fellow man. But as soon as they meet a beggar asking for help on the street, Partridge immediately scolds the guy for trying to get cash from passersby.
Tom laughs at Partridge's double standard: he preaches good Christian virtues when it keeps him out of the army, but he quickly drops his Christian talk when he might have to give alms to the poor. This contradiction in Partridge's thinking is really obvious, which, okay, doesn't make his hypocrisy right. But it does make his double-talk pretty laughable, which is what Tom Jones (as a satiric comedy) is all about.