The tenant pays renter's insurance, which insures the renter's personal possessions.
The owner pays homeowner's insurance, which insures the structure.
Both insurances may (and should) cover liability.
However, since you're asking the question as a renter, I'm guessing you've been asked (or are concerned about being asked) to pay the owner's insurance. First--and check with an insurance agent on this--you don't even have the standing to buy insurance covering the structure. On the other hand, an owner could put something into a lease requiring you to pay a certain dollar figure--say $40 a month--that the owner says would be used for homeowner's insurance. That'd be odd, but possible.
If you can be more specific, maybe we can help some more.