normally at the end of the rent/lease period. At that time, if you decide not to exercise your option to buy the property, all of the monies that have been collected and set aside by the owner revert to him/her, just as if they were always a part fo the normal rent payments.
Should you decide to go ahead with the purchase, the monies that have been set aside are contributed by the owner at closing, just as if you had been putting that money in a bank all along (albeit a bank that was paying you no interest for that time). Depending upon the agreement that you negotiated up front, that could amount to hundreds or even thousands of dollars towards the purchase price of the house.
Some rent-to-own deals require that the purchase agreement be signed at the time that the rental agreement is signed. Those papers are then put away until the anniversary date and either executed as a sale or rendered invalid by the renters decision not to proceed. If nothing else, the rent-to-own purchase agreement gives you the renter the first right of refusal on the house upon the anniversary date.
Every rent-to-own contract is a custom agreement between the owner(s) and the renter(s). You should seek good real estate advice and perhaps even legal advice before signing any agreement.