Compose is to make up a whole, and comprise is to contain parts.  Poodles compose the dog class because the class comprises poodles. The parts compose the whole, and the whole comprises the parts. Confused? Everybody else is!

Compose means to make up a whole (it also means to create, but that's not the definition that troubles people). Mention the parts first, then the whole, as in "four rooms compose our small house." The whole can be first if you say composed of, as in "our house is composed of four rooms." Check it out:

Fresh pītal flowers composed the royal throne, Golden blossoms raised the state umbrella. (Vidyapati Thakura)

The nest is composed of dry grass and lined with finer fibres. (Rev. C.A. Johns)

Comprise means made up of, contain, include. When you use comprise, mention the whole thing first, then the parts, as in "the house comprises four rooms." It doesn't need an of after it. (That's the other word.) Here are some examples of the "whole" comprising the "parts:"

The stories comprised in this volume deserve to be widely known and appreciated. (Williams Andrews)

"Earth" comprises clay, mud, ochre, marl, and "peculiar earths generally." (Georgius Agricola)  

The Penn State police force comprises 46 armed officers. (Slate)

Often comprise shows up in the phrase composed of, but really, comprise does not like of to follow it around. Usage maven Bryan Garner points out that this error is at stage 4 of his language change index: many are mixing up the words so much that the language is changing, but it hasn't changed yet. Remember that those fluffy poodles compose the dog class because the class comprises eager pups. Woof.