Character skill and attribute development in FFX is conducted exclusively on the Sphere Grid. At the very beginning, each of your characters has its own place in the grid, The Sphere Grid is, as it name says, a board with interconnected tiles that contain spheres. You can move your character across the board through the game, the more you advance, the more abilities your character will have. You have to be aware that depending on the path you choose, your character will obtain certain abilities, and of course the most powerful abilities and spells are hidden in the huge Sphere Board of FFX.
If you think you can get the most powerful spells easily with the help of the Complete Sphere Board you are wrong. There are paths that are "locked", which means that you have to unlock them with the "Lv. 1 Key Sphere", "Lv. 2 Key Sphere", "Lv. 3 Key Sphere" and "Lv. 4Key Sphere". Of course, it won't be easy to obtain some spells like Ultima.
Navigating through the Sphere Board not only grants you extra spells. It can also give you more HP, more MP, increase your speed, defense, attack, etc... depending on the color of the sphere on the sphere board.