I cut arm yesterday. is a small cut but is deep and it may need stiches. what happens if i don't get stiches? will it heal itself?

It probably: Will heal, but how well and did you cut anything important like tendons or nerves. I would at least have a doctor evaluate it for your own good. Also, are you up to date on the tetanus?

Puncture wound?: May need stitches if wound depth > 1/4”, wound edges are gaping or jagged or if fat / muscle is visibly extending from wound. If you can’t stop bleeding that requires medical intervention. See: http://www.mayoclinic.org/first-aid/first-aid-cuts/basics/art-20056711. It may or may not heal properly. When was last tetanus shot?