There are two different things meant by 'rush' One is speeding up production of a unit/building, you can pop rush (use drafting or slavery to complete it under nationhood or slavery), cash rush (spend money to complete it under universal suffrage), or chop rush (chop down forests for bonus hammers).
The other is quickly getting to a military technology, then making a lot (or all) of that unit to win a war quickly. An axe rush involves connecting bronze with your second or third city, then building a ton of axemen to take over a neighbor before they can do much. A cavalry rush is about getting to military tradition and gunpowder then flooding an opponent with cavalry before they have a real counter.