There are two general types of concealer: Liquid concealer and stick/cake concealer.
. The liquid one is light and, well, liquidy in consistency and usually used for under eye blue/black circles. It usually has a light peach tone to it as peach is opposite to blue on the colour wheel so it helps to conceal the blue under the eyes.This product should be a shade lighter than your skin tone too. A good example,of a drugstore one of these is Rimmel Match perfect concealer. I actually use the MAC Pro Longwear Concealer. I have a review of this product here.
The stick/cake one is usually creamier in consistency and is used predominately to cover blemishes. I would never use this for the under eyes as it can settle into fine lines and makes them more visible. I use the No7 stick one it is good but I don't use it very much as I don't really get blemishes.
There are also coloured concealers if it is redness you are looking to cover then a green concealer is the best and as I said for bluish areas peach is the best.
General tips for applying concealer:
Moisturise!!!! Seriously, especially when using stick/cake concealer. It can be very drying.
Use a light hand, don't put too much on initially, you can always add more but it's harder to take away.
Use clean fingers (best to use the ring finger as it doesn't put too much pressure on say the delicate under eye area.)to tap it in to the skin rather than rub it in because the heat of your fingers will help it settle better.
Make sure that the colour is right for you.
Remember blue under eyes require peach tones and redness needs green based tones.
If anyone has any recommendations of concealers or have any opinions on any of the products mentioned please let me know.