You need seed beads or delica beads (a type of seed bead that is uniformly sized), beading needles, and nylon beading thread. If you want to do loomwork, you also need a beading loom.
Beads
Seed Bead Available in an amazing array of colors, finishes, and styles, they are most often used in bead weaving and loomwork. Larger sizes are smaller, with 11/0 being the most common.
Delica beads A variety of seed bead that is small and perfectly cylindrical, so they easily snap into place in peyote stitch and brick stitch beadweaving. They also have large wholes for their size, making it easier to pass multiple strands of thread through each bead. Delica is actually a brand name, made by Miyuki--the generic term is "cylinder bead".
Tools
Needles For regular stringing, twisted-wire needles work well as the large eye closes to fit through the beads. For seed beads, use beading needles. These resemble sewing needles but are usually longer and thinner. The larger the number, the thinner the needle (to match sizes of seed beads).
Bead Loom A bead loom is similar to a regular fabric loom, except with beads strung on the weft (the thread that passes over and under the warp threads). Typically one bead is positioned between each warp thread, and then the weft is threaded around the final warp and back through the beads to secure them in place. Loomwork is a beadweaving technique that uses a bead loom.
Stiff Stuff Stiff Stuff is A brand name interfacing made by Lacy's, useful as a beading foundation for bead embroidery. Stiff Stuff can be dyed or colored with fabric markers to match your piece so the background is less noticeable.