It can vary, depending on who you're dealing with, and what property you're interested in. Properties can be offered for rent directly by property owner (the landlord) themselves, or sometimes the owner hires a property management company.
A property management company is likely going to have a strict set guidelines and a more rigorous screening process in place. They'll ask you to complete an application, then they'll do a background check, check your credit scores, check references, etc. Owners/Landlords may simply ask for some references and do a background check and that's it. Everyone will want to know what you make and that you have a good history of paying your rent or bills on time. If you've never leased a place before, you may have to get a co-signer. (like a parent or friend). They'll want money up front in the form of a deposit (that you should get back at the end of the lease).
Beware of the scams out there. You may find listings here on Trulia or other sites like Craigslist that sound too good to be true. When you submit an inquiry, you may get a long invovled email back saying they (owner/landlord) is out of the country, and they want you to wire them $$ and they'l mail you the keys. That is a SCAM. Only deal with landlords or property managers in person. Make sure they are indeed the owners or the property managers (ask for proof).