With an option, you agree to rent a house for say $1500/month instead of the $1000/month rate it would normally go for. The extra $500/mo goes toward your equity, and if you do choose to buy in the agreed time period for the agreed price (at time of moving in, not at a later time — e.g. agree to $120,000 1 year from now), the additional $500/mo goes toward the purchase price. If you choose not to buy, then you just lose that $500/mo.
The initial option contract contains all of the terms of the deal: the monthly rent, the amount that goes toward the purchase if the option is exercised, and the purchase price if the option is exercised.
A lease purchase is similar except it’s not optional to purchase at the specified future date.
These two purchase types are most often sought after by purchasers without access to traditional mortgage financing (e.g. credit issues that are expected to be resolved within 3-6+ months).