As Farhan noted, the rules are essentially the same as if you buy a plane to provide instruction in it.
Inspections and Maintenance
In addition to your required annual inspection you will also be required to complete a 100-hour inspection every 100-hours of operation.
You will need a commercial insurance policy which allows you to rent the aircraft.
This policy covers the aircraft, so you will also need (or at least want) general liability insurance and all the other trimmings that come with operating a business.
This is a broad category. There's no way we can cover it all. You will however need to consider:
Maintaining an adequate stock of "service spares" to maintain the aircraft.
Tires, light bulbs, oil, oil filters, 5606 hydraulic fluid, etc. - As an individual owner you may be OK with being down for a week waiting for parts, but as a rental operation that will lose you customers.
Having a relationship with a mechanic to do the maintenance
Owner-Performed Preventive Maintenance goes out the window when the aircraft becomes a rental - you may still be able to assist your mechanic, but you can no longer do & sign-off on things like oil changes.
Vetting your renters
Rental aircraft at an FBO/Flight School live a hard life. If you like your plane and want it to stay in good condition you'll need to vet your renters a bit more thoroughly than "You have a pilot's license and you went around the pattern without hitting anything" - Find the kind of pilots who will police their trash from the cabin, lean the mixture appropriately, etc.