If you want to market your home for rent on the MLS, a Realtor could do that, but they typically charge a commission on one year's rent (commission amounts vary from neighborhood to neighborhood, and are negotiable). However, this fee is not paid until they find a tenant, get the lease paperwork in place, and receive the tenant's deposit and 1st month's rent. It is from the deposit and rent that the landlord will deduct their commission. Typically the Realtor's obligations will be fulfilled upon successfully getting a tenant in place in your rental property, so the landlord will be responsible for any further dealings with the tenant (receiving rent checks, handling repairs, evictions, etc).
If you need full service support, so that someone else can handle those 9:00 pm calls on a Sunday because the toilet overflowed, then a property manager would be the way to go. They will take care of everything, and you'll receive the checks each month. They do charge more, and you will want to carefully interview the managers to make sure they are experienced and qualified to handle any issues. A property manager shines when they successfully deal with problem tenants.
I hope this helps! If you have any followup questions, I'd be happy to answer them.