Yes. All beds are designed and built to fit standard mattresses according to the stated sizes. If you choose a twin bed a twin mattress will fit, if you choose a queen bed a queen mattress will fit, etc. etc.
NOTE ABOUT SOME BUNK BEDS: Twin over Queen or Full over Queen Bunk Beds will use an extra long twin or an extra log full/double mattress on the top bunk due to the queen mattress being longer.