Needle felting is done with special, barbed felting needles which push the layers of wool together without water.
Just as in traditional felting, you need wool roving. Other fibers such as llama, alpaca, and angora (rabbit) are more expensive but give a different feel.
Be extra careful. Felting needles are much sharper than normal needles with barbs along the sides. They come in different sizes for finer or looser felting work. Many felting needles come as a kind of multi-needle tool with a handle. You can find many here.
You will also need a spongy, foam mat to felt on (the needle will poke into this instead of your table!). You can pick up a plain mat at a craft store or purchase a specialty mat for felting, which looks like a stiff, flat brush with a plastic base. This mat makes removing your felted piece a breeze and no foam is stuck in your piece.
Just as in traditional felting, you put down a layer of wool in one direction, and another layer in the perpendicular direction, on top of your foam mat. You can put another layer down in the original direction for three layers and a thicker result.
Poke your felting needle (or multi-needle tool) into the wool rapidly over and over again, moving randomly over the piece. You only need to go about ¼ inch into the foam, so not too hard! It’s important to keep your needle straight up and down, not at an angle, or you risk breaking the needle.
If a piece is too thin in places, add a bit more wool to that area and use the same technique to poke it down.
When it’s pretty well matted down, lift the felt off the mat, turn it over, and repeat the process. You may have to flip your piece over several times.
To smooth out the “fuzzies” on the side of your piece, put on rubber finger protectors, and use a single felting needle to push them into the sides and back.
For a bit more clarification on the general idea of needle felting, check out this video demonstration from Debb41. You'll see how the wool fibers start to mat down as the needle is applied.