Your under-eye concealer probably came with an applicator sponge. That one will work, but it's not ideal. First, many of those sponges contain Latex, which can be rough on sensitive skin. Also, they'll fall apart after a couple of washes, which are crucial (more on that in a minute).
There are lots of fancy sponges available, but the single sponge you really need for smoothing and blending your liquid or cream under-eye concealer and shadow primer is a small, Latex-free sponge. You'll know the right one by its rounded edges, which make it easy to get around the contours of your face without streaking.
Washing any makeup sponge is a must. A weekly flush with gentle soap and water will be enough to keep your sponge in top form and bacteria-free.
If you follow your eye primers with anything powdered, and most women do, you'll want a few good brushes. The key is using the right brush for the right powder.