Every phone has a different way of rooting, so the best way to find out is to search for methods specific to your make, model, and even version (I.e. A Sprint method may not work with a Verizon phone, and those may not work with non-carrier phones). Chances are you'll wind up over on the XDA forum for a lot of info. Also keep in mind not every phone has a root method. Less popular phones may not have enough demand for people to find an exploit, which can take a lot of time and effort. Above all, rooting involves a fair bit of risk. If not done correctly, you can brick your phone. So do a lot of research before even considering it, making sure you understand all the steps for your specific phone. You'll also want to research ways of fixing your phone if possible, in case you do screw up the rooting process.
Rooting allows greater control over the phone. Rooting itself, when done correctly, doesn't make any changes other than giving you more access. What you do with that access is up to you. Some things I've done with it is allow for lower than stock screen brightness without needing a "screen filter" app (actually lowering the backlight more than normal), undervolting the processor to increase battery life, and to purposely break the carrier OS update ability so I wouldn't have one forced on me unless I was ready. Other people go more in depth, even going so far as to essentially replace the OS with a custom ROM.