Renters Insurance is home insurance when you don't own the house / apartment you live in. As its title suggest, you are renting and this insurance protects you from two things like homeowners; your Contents and Liability.
You are renting your apartment and hopefully you have some furniture, a bed, maybe a laptop, phone, clothes, ext. If these items are damaged, you may be covered under your renters policy.
You are renting an apartment in Midtown Manhattan and left the water on by accident. When you come home you find water all on the floor and the tenants on the three floors below have water in their units also. This is where Renters Insurance comes in. Like Homeowners Insurance, you are also protected for some lawsuits and bodily injury.