The textbook definition of a parasite is an organism that spends at least part of its life cycle in or on another organism, harming it in a way that benefits the parasite. The organism the parasite feeds on is called the host. Most parasites are smaller than the host organism - usually much smaller, as with fleas and ticks - and reproduce more quickly.
While there are many kinds of parasites, all fit into one of two fundamental types of parasitism: endoparasites, which exist inside a host's body; and ectoparasites, which exist outside a host's body. There are many kinds of parasites within each of those categories as well.