Typically, you pay to apply to rent the property.
You pay a security deposit upfront if approved and for pets a pet deposit.
Prior to move-in you'll pay the first month's rent or some fraction of it.
While you live there, you'll pay for utilities, like electricity, water, gas; but also any optional services like cable TV, Internet, alarm monitoring, phone etc.
You'll also pay for lawn service, and if there's a pool, pool service.
Some places the owner will pay for some of these items, but typically you'll be on the hook for all of them.
If something breaks, you may be on the hook for all or some of the repair bill. Usually, the owner pays for plumbing, water heater and HVAC repairs, unless they have been abused.
Your security deposit covers any repairs when you move out, but not the final month's rent, plus cleaning to put it into showing condition.