Well I think as a Sith I'd be looking for someone physically strong, and trainable, but they'd have to be weak and easy to manipulate to do my bidding. As a Jedi, I'd probably look for someone who strongly stood for their beliefs, and was willing to give up certain things in life to follow the ways of the Jedi.
If I was a Jedi Knight looking for an apprentice I think I would choose someone who is adventurous, and yet obedient and loyal. Someone I can trust and who can trust me. Someone who will respect and honor the Jedi council and its decisions. Obviously I would like skill, but not too much to make him/her arrogant. I would like room for improvement. But I think most of all I would like someone who can teach me as well. Of course, if I were a Sith, I would look for someone completely different. Someone who is strong and powerful. Someone arrogant, so that they are weak inside and that I could bend them to do my bidding, whatever it was. Someone completely and utterly evil. Not that I would ever be a Sith though. The Jedi rule. For both however, I would prefer another human and they should be strong in the force.
I would want a Jedi apprentice and not a Sith. It wouldn't matter what species they would be because looks don't matter. It's what's inside the person that matters. I would want my apprentice to have a good heart like myself, be willing to think for themselves and be brave not being afraid to stand up for what they believe in and willing to give their life if need be.