Carry on bags are supposed to fit in a metal frame at the gate; I've never actually seen anybody's bag measured, but I've heard it happens.
I've used a shopping bag on wheels; a rolling backpack; and a standard small suitcase (all at different times, of course!)
Check your airline's website for specifications; some are stricter than others; and if you use suitcases, PLEASE put them in the overhead compartment wheels-first. This way, three suitcases plus some other stuff maybe can fit. Passengers who put the bags in sideways prevent others from carrying on suitcases.
You are also each entitled one smaller, personal item - a briefcase, computer case, purse, etc. That's it. One carry on and one personal item per passenger.