"Support Pokemon" usually refers to Pokemon that aren't supposed to knock out opponents, and instead use their moves or abilities to help their teammates. "Support moves" usually refers to the moves that are used to help teammates. The best examples of support abilities include condition setters, like drizzle or psychic surge. Examples of support moves (in singles) include hazard setting moves, switching moves (u-turn, baton pass, etc.), knock off, condition-inflicting moves (taunt, toxic, etc.), hazard removing moves, trick room, stat-resetting moves (haze, whirlwind, etc.), memento, healing moves (wish, aromatherapy, etc.), and screens. I can't come up with such a list for doubles, but I hope the answer is good enough as is.