Utility patent is for invention. It protects the way how an article is used or worked. It forbids others (individuals or companies) from making selling or using the same invention without permission of the real owner.
Design patent is for design. It protects the way how an article looks like. A design patent allows an inventor to protect the original shape or surface ornamentation of a useful manufactured article such as industrial design, ornamental design or computer icons etc.
Plant patent is for newly invented plants. These plants must be reproduced by other means rather than seeds, for example by drafting, budding or inarching.