From the wiki:

• DIY – Do it yourself. This means all you pay for is the stall space to keep another horse. The boarding center will not have any care benefit to you. You will still have to manually feed/water/groom your horse.
• Part- Part care includes feed and water. Generally, part care costs more than DIY care as the boarding center must cover feed and water costs. If you only want to have to groom horses this option is for you!
• Full – Full care includes feed, water, and groom. Full care typically costs the most but ensures your horse is always fully cared for. If you wish to not have to worry about daily care, this boarding option is recommended.