What do we gain by running a node and what does it accomplish for the Monero network?
Monero has two primary traits that set it apart from Bitcoin (and alt coins that are meant as currency): fungibility and privacy. The two go hand in hand, in the case of Monero, since the same technology that allows for privacy also ensures fungibility.
Many would say that fungibility is the most important trait, because privacy isn't really as important (it potentially loses its impact) if the currency isn't fungible. For example, if you could privately/anonymously send coins known to be tainted, you would stay out of "trouble," but those coins would still potentially hold less purchasing power than other, untainted coins. Many would consider that a problem. Well, that can't really happen with Monero. Monero is fungible, and it's true whether you're running your own node or not.
With fungibility a given, now we can talk about privacy. If you hold coins on an exchange, the exchange can see your balance and your activity (and they ultimately control your coins, in most cases, but that's beside the point). If you hold your funds in a hosted wallet or web wallet, your balance and activity (deposits anyway) can be seen by the host. If you run your own node, however, you now have a chance at full privacy. Assuming your computer is free from malware, you are the only one that has a clue about your wallet history. If you value your privacy to that extent, then that's what you get by running your own node.
To answer your second question, I really must parrot the existing answer, that you would be making the network more decentralized and distributed.
By running a Monero node, you gain access to a fungible currency (albeit with large price volatility). It is like Bitcoin, but better (subjective, of course, there are pros and cons). You can send and receive monero privately (privacy/fungibiity are the twinned major changes from Bitcoin).
The network gains decentralization too: it becomes harder to perform segregation/partitioning attacks. If you mine, you make it harder to an attacker to 51% the network (and you also get block rewards, albeit infrequently).