Deadheading is the process by which a gardener removes the dead flowers from the plant by pinching or cutting them off of the stem. Deadheading is done for a variety of reasons such as making your garden look more fresh and vibrant by getting rid of the brown and crunchy flowers, encouraging the plant to bloom again, and preventing the plant from going to seed.
There are a couple things to consider before you decide to deadhead. If you want to gather seed from your garden or want seed from the existing plants to germinate and fill in any clear spots, deadheading isn't the way to go as it will prevent the plant from producing seed. If your overriding concern is having a lush, green, healthy looking garden then you will almost certainly want to deadhead.
There are a couple of different ways to deadhead flowers. One is to pinch below the dead flower at either the first set of lateral leaves or the next flower bud if there is one, and pull off the dead tissue. If you're deadheading a plant, like a lily, that doesn't have lateral leaves then you remove the stem as close to the ground as possible.
Another method, and the one I personally prefer, is to use long-bladed pruning shears or scissors. I find that they make a cleaner cut and cause less tissue damage to the plant. However, it's a lot easier to make a mistake and remove a healthy leaf, stem, or bloom when you're using pruners or scissors so make sure that you're paying close attention.
Now, no two species of plants are the same and there's no real deadheading "rule of thumb" that covers all different kinds of plants and I don"t have the room here to list them all. If you're not 100% sure it always pays to "Google" whether you should deadhead a particular cultivar or species of plant.