In addition to soft corals (AKA octocorals), there are some other critters that are relatively easy. Mushroom corals (AKA mushroom anenomes, corallimorphs, or corallimorpharians) are hardy, often tolerant of lower light than hard corals, and usually quite colorful. Zoanthids, palythoas, and their relatives (AKA zoos, zoas, palys, or just "polyps") are also good starters, and come in an incredible variety of color combinations.
There are three main things that make hard corals (AKA hexacorals) more difficult than soft corals, especially for beginners. First, they usually need more intense light than soft corals, mushrooms, and zoanthids. Second, they sometimes need stronger water flow than the other animals. Third, they need high water quality, and they don't tolerate much variation in the water quality.
These qualities make hard corals tough for newcomers, who often have lower-powered lights and fewer powerheads, are less likely to own or have access to a Reverse Osmosis-DeIonized (RO/DI) water system, and have less experience at maintaining water quality. Many newcomers also tend to start with smaller tanks, and the quality of a small volume of water will change more quickly than a larger volume. It's counter-intuitive, but in this hobby, smaller systems can be harder to maintain than larger ones.
Saltwater hobbyists have divided the hard corals into Large Polyped Stony (LPS) and Soft Polyped Stony (SPS) corals. This distinction doesn't come from marine biology, but from practical reef-keeping experience. In general, LPS corals are harder to take care of than soft corals, zoos and mushrooms, and SPS corals are more difficult than LPS corals. A good LPS coral for a relative beginner would be Duncanopsammia axifuga (AKA Duncans, sometimes Duncanos), but like mcfeefer, I would advise starting with softies, shrooms, or zoas before trying your first LPS.
Edit: Another quick tip - Leave anenomes for later, too. Many people start with clownfish in their first salwater tank, and they want their fish to have an anenome to live with. Unfortunately, it's much harder to keep anenomes alive than clownfish. Luckily, clownfish are very adaptable, and will happily use many different kinds of soft corals, and several types of LPS, as substitute anenomes.