There are a couple of different concealer types out there, and they're designed to do different jobs.
These are you correcting concealers, your green and yellow colours. It's worth having both of these in your kit. Salmon will do a great job of hiding any dark circles under your eyes, and green will help neutralise any redness from pesky spots that pop up. If these come as a stick, I like to use my fingers to blend and pat the concealer into the skin.
Most commonly used under the eyes, illuminating concealers reflect light, making you seem more awake and blurring any fine lines. These are also good to use just under the eyebrow for and instant eye lift, and on your cupids bow to make your lips look a little bigger.
These are the most common type of concealers. You usually get them either a squeezy tube, or in a bottle with an applicator wand. The opacity on these varies greatly, but they work well for covering most issues, including up eye circles and blemishes. You can either use a flat concealer brush or a fluffy brush to apply the concealer, depending on how much coverage you want, and then gently pat it in. This keeps the concealer just where you needed, rather than spreading it out and sheering it down.
You get these in a pan. They tend to have a higher coverage than other concealers, so do well at covering blemishes. Because of their thicker texture, they do tend to sit in fine lines under the eyes. These are another product that I like to apply with my finger, as the heat helps soften them and make them easier to blend in.