A free lease is basically as if you've sold the horse. The person leasing pays everything, from board to vet bills to lessons... but you still own the horse. Both would sign a contract as to the terms and anything extra. So you would have the final say in pretty much anything to do with your horse.
A paid lease (full) would be the same thing as a free lease (the person leasing pays everything) but they also pay you a monthly fee for the horse.
A half lease is usually where each person gets 3 days with the horse (usually a lesson and 2 free rides or 2 lessons and 1 free ride). In this case the total cost of keeping the horse per month would be split evenly by the two people leasing.
As for convincing your mom, I've never really had to do that but maybe just explain to her that you'd really like to advance in your riding and have fun again but that's not something you can do with this particular horse. Chances are he'll be perfect for someone else and both you and he will be happier in the situation.
Hope this helped.
The essential joy of being with horses is that it brings us in contact with the rare elements of grace, beauty, spirit, and fire. ~Sharon Ralls Lemon