Profile is the shape and height of the keycap, row by row. It changes how far your fingers travel and how the board sounds. It does not change whether a set fits — that is layout coverage, which is a different question.
The four you will meet most
Cherry is low and sculpted; the most common choice and the easiest to adapt to. OEM is slightly taller with the same sculpting. SA is tall and heavily sculpted — distinctive, louder, and an adjustment. XDA is uniform height across all rows, which some people love and others never get used to.
How to choose
If you have never changed profile, start with Cherry or OEM. If you already know you like a tall, scooped feel, SA is worth the adjustment period. Uniform profiles like XDA suit people who move keys around often.