Ganya is an overly vain and ambitious functionary in General Epanchin's office. After a failed attempt to marry Nastasya for money although he actually despises her, Ganya makes a failed attempt to marry Aglaya for love.
One of the novel's running themes is the idea of the person who is an outlier. Think about it—most of the main characters are just off in one way or another—and of course none more so than Myshkin himself. But although these characters spend their lives on the periphery, they are constantly trying to get into normal life somehow.
Nastasya keeps trying to convince herself to just marry Myshkin already, Aglaya tries hard to make her family happy by not being as unconventional as she wants to be, and Myshkin doesn't particularly realize that he is a nut—but if he did, it's clear he would want to change in order to make people less uncomfortable. On the other hand, though, we have Ganya, a totally average person whose primary ambition is to become somehow unique, unusual, and hopefully awesome.
It's interesting that we learn most of this in a long-winded diss from the narrator, who first writes in general about mediocre people who will never amount to much, and then zeroes in on Ganya himself—mediocrity personified, who wants sadly and pathetically to be something more. Check out the disdain dripping from the narrator's pen:
Our friend, Ganya, belonged to the other class—to the "much cleverer" persons, though he was from head to foot permeated and saturated with the longing to be original. This class, as I have said above, is far less happy. For the "clever commonplace" person, though he may possibly imagine himself a man of genius and originality, none the less has within his heart the deathless worm of suspicion and doubt; and this doubt sometimes brings a clever man to despair. (As a rule, however, nothing tragic happens;—his liver becomes a little damaged in the course of time, nothing more serious. Such men do not give up their aspirations after originality without a severe struggle,—and there have been men who, though good fellows in themselves, and even benefactors to humanity, have sunk to the level of base criminals for the sake of originality). (4.1.11)
It's really all there in those quotation marks around the words "much cleverer"—what a slap in Ganya's face. And really, the tragedy of the character is that he has the desire to do something outrageous in order to get attention, but he can't actually make himself do any of the things that opportunity presents.
At the start of the novel, we get the sense that he could elope with Aglaya, but he is too tempted by the money he'd get for marrying Nastasya. He could marry Nastasya, but he's too worried about everyone else's opinion of her to pull that trigger. So he mostly just spends the novel becoming more and more bitter about his life and being a good example of someone so pulled in different directions that he just ends up standing still and doing nothing at all.