Most airlines allow one piece of carry-on luggage or “hand baggage” that can fit in the overhead bin, plus a “personal item” (a smaller purse, computer bag, diaper bag, small backpack, etc. which can go under the seat in front of your).
Does purse count as personal item?
A personal item is any type of luggage that fits underneath the plane seat in front of you. Most domestic airlines in the United States require a personal item to be 18” x 14” x 8” or smaller in size. Personal items can include a purse, briefcase, small backpack, laptop bag, weekender or small duffel bag.