Depends on where you cut it and what you can still do with it. If you cut it at the fingertip, the only thing that really matters is infection. Clean it out thoroughly with water and/or iodine to sterilize it a bit, then just wrap it up a bit so that the skin has a chance to heal the flap properly without it being moved around every 10 minutes. It is absolutely certain that you would have hit a nerve (or more likely, several dozen) but this is of little consequence. You'll lose some sensation in the fingertip, but the nerves will eventually grow back.
If you cut the finger closer to the hand but it's still shallow and you can still move the finger completely fine without problems, it's unlikely you damaged a tendon. If it's deeper or if you have pain or difficulty moving your finger, or if blood is coming out in spurts rather than in an ooze, you should get it looked at straight away. If you hit the bone, that's also something you should get looked at.