Sandbox is a gamemode without any enemies, allowing you to take your time and build anything you'd like to, without any restrictions or fear of being attacked. In multiplayer, however, it is possible to attack other players with certain dynamic objects such as catapults and explosive barrels.
In Invasion, formerly known as Survival, your objective is to survive as many waves of Corruptrons as possible. Every few minutes, a wave will begin and a horde of Corruptrons will attempt to destroy your Home Crystal, which you must defend at all costs by building walls and spawning military units.
There is no upper limit of waves. Eventually, you will be overrun, it's inevitable.
In Conquest, you must slowly expand your territory by capturing shards and defending them against enemies, who are also trying to expand. In order to win, you must capture your enemies' Home Crystals. If your Home Crystal is captured, you will be defeated.