On ANY airline*, each ticketed passenger is allowed ONE carry-on that will fit in the overhead compartment - see your specific airline for exact dimensions, keep in mind that any such carry-on should go in wheels-in or wheels-out (vs. sideways) to enable the most passengers possible to fit their luggage - and one personal item such as a purse or laptop case that will fit entirely under the seat in front of you.
*Any airline, really, except that while most airlines allow this carry-on at no cost, Spirit charges; also, you may be required to gate-check the carry-on if too many passengers fill up all the space before you board.