PLEASE don't go the route that everyone seems to be going and get an under eye concealer that is TOO LIGHT. It makes you look funny... especially in photographs.
TRY to match it to your skin coloring.
If you have PURPLE under eye circles, try to find one that has a yellow or warm base to it. You can usually tell by the (W) next to the name of the color.
Dab it. Don't smear it. Concealer will not stay where you want it unless you dab it and keep it there.
Again, TRY to match it to your foundation color, (which HOPEFULLY matches your skin color!).
One thing I like to do with concealer is apply it AFTER my foundation and powder. This way, it stays put a little better. Most people put it on first, and if you do that, when you apply your foundation, you'll move it around, and you'll have to re-apply usually.
So, things to remember about concealer:
1. MATCHY MATCHY
2. DAB IT!
3. APPLY OVER YOUR FOUNDATION
Hope this was helpful.