Rushes are strategies, usually early-game, that use speed, surprise, and numbers to win. They are effective early-game because the enemy will not have had time to set up a proper base and their defensive capability is minimal.
Allies
Vindicator Rush - This tactic is available early on for the Allies. Just build a Power Plant, then an Airbase, purchase the Advanced Aeronautics upgrade from the Top Secret Protocols menu and finally build as many Vindicators as you can and send them to level the enemy. This is ineffective against mass Flak Troopers and against enemies that have Mecha Tengus, although you can strongly minimise the Tengus effectiviness by ordering Appollos to escort your Vindicators. One upgraded Appollo can hold against to 3 Tengus simultanously(!).
Soviets
Flak Troopers and Bears - This tactic uses Flak Troopers and War Bears to quickly decimate an enemy base. The Flak Troopers use their Magnetic Mines on enemy structures and War Bears to ensure no infantry can bother you. If the enemy has managed to set up defenses however, your rush will probably fail.
Empire of the Rising Sun
Dojo Core Rush - A tried-and-tested strategy, the Dojo Core Rush involves rushing Dojo Cores to an enemy base and setting them up inside or nearby the base. Then, you pump out Imperial Warriors and Tankbusters to quickly overwhelm and destroy the enemy infrastructure. While not foolproof, this tactic can destroy or severely cripple an unprepared enemy.