The only "essential" tool is a "barbed" felting needle.
There are a variety of felting needles available.
They are sold by gauge from 36 to 42. The higher the gauge number the finer the needle.
There are two shapes, 3 sided "triangle" and 4 sided "star" shapes.
I use a large needle when I'm working with coarse fiber, a medium needle for medium to fine fiber and roving
and a fine needle for the finest details and soft smooth finishes.
Various brands of needles have differences in the length of the barbed area and the number and position of barbs.
But don't worry. It won't take long to get the hang of choosing the right needle. For example:
If you are working with coarse, loose fiber a 42 gauge triangle needle will slide right through without snagging any fibers.
In the same situation a 36 gauge triangle needle will grab the fibers and start pulling the felt together.
If you are working on a tightly felted figure it will get harder to push the thick needle in.
Eventually you might even break the needle.
But if you switch to a thinner needle it will allow you to keep refining the surface.
Just a hair away from essential is a foam surface to work on. The foam lets you push through the fiber without
breaking a needle in a hard surface or stabbing yourself!
A "spongy" foam like the green foam sold for chair pads, or the anti static foam they ship cameras and lap tops in
both work fine. Do NOT use Styrofoam or floral foam. They will break your needles and shred into your felt.
To keep your work clean you will need to switch pieces of foam when you switch colors or cover the foam with a
thin piece of cloth.
You might also like to have some other stuff. Carders for blending fibers, beads for eyes, thread and needles to attach the beads, a scale and stuff to dye fiber.