Capellini are thin rods of pasta slightly thicker and otherwise very similar to angel hair pasta (Capelli d'angelo). Each capellini rod is about 1.2 - 1.4 mm in diameter which is less than 1/16 of an inch.1 The pasta is commonly sold in flat boxes like spaghetti. Don't confuse this pasta with cappelletti which is a small stuffed pasta.
On an interesting side note, cookbook author Giuliano Bugialli mentions that many of these types of pasta rods were originally distinguished by their length and perhaps to a lesser degree than their width. With the onset of mass production the pastas were cut the same length to fit into standard box sizes so eventually became almost identical products.
1Source: Culinaria Italy
Use capellini in brothy soups or simply with butter and a little Parmesan cheese. A classic preparation would be capellini with a light tomato sauce with fresh basil. This pasta cooks very quickly in about 3 minutes. Don't overcook!
If your local grocery has a well-stocked pasta section you should be able to find it. If not you can also purchase it online at Amazon.com, or just buy angel hair pasta which may be more easily found.