Utility patents are most common and offer the longest protection. A utility patent may be granted to whoever "any new and useful process, machine, manufacture, or composition of matter, or any new and useful improvement thereof." 35 U.S.C. §101. The primary purpose of a utility patent is to protect the function of your invention.
Design patents are less common and offer shorter protection, but are also cheaper and take less time to obtain. A design patent may be granted to whoever "invents any new, original and ornamental design for an article of manufacture." 35 U.S.C. §171. The primary purpose of a design patent is to protect the appearance of your invention. Design patents are therefore more specific, and vulnerable to infringement by similar products, that do similar things, but only look slightly different. It is possible to get both a design and utility patent on the same invention, but tricky and full of pitfalls.
Plant patents may be granted to whoever "invents or discovers and asexually reproduces any distinct and new variety of plant, including cultivated sports, mutants, hybrids, and newly found seedlings, other than a tuber propagated plant or a plant found in an uncultivated state." 35 U.S.C. §161. There are additional requirements for plant patents, and certain types of plants, such as potatoes, are not patentable. The two key requirements specific to plant patents are (1) the plant can be reproduced without the use of seeds (asexually), and (2) the plant has not existed in nature. Even though plants produced by seeds cannot be patented, they can be protected under the Plant Variety Protection Act, which is administered by the United States Department of Agriculture (USDA).