Most concealer problems come from treating it as one product with one job. It has three, and they want different things.
Under the eyes
Shade: one step lighter than your foundation. Not two — that produces the pale patches you see in daylight.
Placement: an inverted triangle from the inner corner down toward the cheek, not a semicircle following the shadow. The triangle brightens the whole area; the semicircle emphasises exactly what you were hiding.
Technique: press with a fingertip or a damp sponge. Do not rub — the skin there is thin, and dragging both creases the product and is bad for the skin.
Setting: a very light dusting of powder, only if the product moves on you. Too much powder is what makes under-eye concealer look dry by lunchtime.
On blemishes
Shade: matching your foundation exactly. Lighter concealer on a spot creates a pale dot, which draws more attention than the spot did.
Placement: on the mark itself, not around it.
Technique: apply after your base, not before — foundation applied on top wipes most of it off. A small brush, patted rather than blended outward.
On redness
Shade: a colour-correcting product if the redness is strong, otherwise a foundation-matched concealer.
Green-toned correctors work because green sits opposite red on the colour wheel, but they need a thin layer with base over the top. Used heavily they are worse than the redness.
The three mistakes
Too much. Concealer is the easiest product to overapply and the most obvious when overapplied. Start with less than you think.
Wrong order. Base first, then concealer. Doing it the other way undoes the concealer.
Not enough blending at the edges. The centre can be opaque; the edges must disappear. Every visible edge reads as makeup.
When concealer is the wrong answer
Very dark under-eye circles, dryness and flaking cannot be fixed with product — heavy concealer sits in the texture and looks worse than bare skin. Sleep, hydration and eye cream do more, and a lighter hand plus accepting the result usually looks better than a thick layer.
For choosing a base to match it to, see the foundation guide.