In the US, renter's insurance typically covers the tenant's personal property and can also include liability coverage that could cover accidental damage to the rented property caused by tenant or guest of tenant. The landlord will typically have insurance that just covers basically the unit as it was when you occupied it, so in case of a fire that burned the rental to the ground their policy would help them recover but would do nothing for your lost possessions.
Typically you will put down a damage deposit equal to one month's rent at lease onset that the landlord will use to cover damages (caused by you or your guests) in excess of common wear/tear. That damage deposit does not free you from liability if the damages caused exceed the amount deposited, but renter's insurance can cover these accidental damages.
It is generally a good idea to carry renters insurance and some landlords require it, the policies are relatively inexpensive. Just be sure to understand what the specific policies you consider do/don't cover.