Renters Insurance is like homeowners insurance for renters. Some apartment complexes in town require tenants to carry a renters insurance policy, but don't despair—it's pretty cheap. Many (most? all?) auto insurance companies give hefty discounts on your auto policy if you pick up a second policy—homeowners, renters, whatever. You may even end up with net savings, actually saving more on your auto policy than you pay for your renters policy (State Farm Insurance example: with a two-car policy, a base-line $80/yr. renters policy knocked more than $100/yr. off one of the car policies).
Renters insurance is different from homeowners' insurance in a few ways. Policies usually cover damage to your personal belongings and liability for injury in your rental property. The policy doesn't cover damage to the building itself—your landlord should have a policy for that.