Magic in Chrono Cross are generally divided into six different categories, according to color. The colors are red (fire), blue (water), green (earth), yellow (wind), white (holy) and black (evil, I guess). Each character, monster and opponent is of one element, i.e. one of the six colors. This means that if a spell of opposing color (e.g. black on white) to the character is used on him, it will do maximum damage. For example, Serge is of white element, and so black spells will be highly effective on him. If a spell of the same element is used on the other hand, it will do minimal damage.
In order to use a spell, you need to equip it in your magic slots. Spells can be obtained by buying them at shops, finding them in treasure chests or defeating enemies. There are two types of spells, one use only and re-usable spells. One use spells have a 1X symbol next to the name of the spell, and can only be used once. These spells are usually recover spells. Re-usable spells, on the other hand, will not disappear after use. However, these spells can only be used once per battle, unless you equip more than one of the spell. You can equip five one use spells in one slot, where as you can only equip one re-usable spell per slot.
Each spell has a number, which is usually something like 1 plus or minus 7 or 4 plus and minus 4. Each vertical bar represents one level of magic, the most left hand vertical bar being Level 1. The number before the plus and minus sign is the preferred level for the spell to be equipped, and also the level that spell will be equipped if you use auto-equip. The number after the plus and minus sign is the range that spell can be equipped. For example, if a spell has 6 plus or minus 2, it can be equipped anywhere between Levels 4 to 8, but it is preferred you equip it at Level 4. If you equip the spell higher than the preferred Level, that spell will be more effective, and vice versa. A character can equip any color spell, even though the character is mainly of one element. This only means that spells of the same color as the character will be more effective, and vice versa.
In order to cast a spell of a certain level, you'll need to gain that level first. Levels are gained by spending Stamina points for attack. Say, if you use 3.0 points, you will gain three levels of magic. Only successful attacks will gain the levels, missed attacks will gain none. This means some kind of a strategy needs to be implemented, as you can put a spell on a higher level to make it effective, but you'll need to spend more Stamina points to gain the required level.
Summons can be used in Chrono Cross, but their usage comes with some requirements. You'll need to spend a star for each summon spell you use. Stars are gained by defeating most bosses. Summons are also like spells, they need to be equipped. However, summons can only be equipped to a character who is of the same elemental color as the summon, which is an exception to the normal magic equipping. Also, if you notice, there is a field effect bar at the top left side of the screen. Every spell's color that is cast will be recorded on this bar, with the most recent color filling the inner one, and pushing the others back in the process. In order to use a summon, you'll need to have all three bars filled with the color of the summon. For example, Salamander, a red summon, requires the bar to be filled fully with red. This is easier said than done, as your opponent's spells will also be counted into this bar, and might disrupt your summoning plans. However, spells that turn the field effect into a singular color are also readily available.