An analogy compares two unlike things to illustrate common elements of both. An analogy essay is an extended analogy, which explains one thing in considerable depth by comparing it to another. Analogy essays can be used to discuss nearly anything, as long as the writer can find a comparison that fits.
Come up with an analogy. One half of the analogy is the thing being explained, while the other half is the explainer. For example, if you said growing up is like learning to ride a bike, you would be explaining something complex and subtle (growing up) in terms of something simple that your audience will be familiar with (riding a bike.)
Draw a vertical line down the middle of a piece of paper to divide it in half. On one half, write characteristics of the explainer, and on the other half, the explained. Try to match up the characteristics. For example, training wheels might be similar to having to have lots of supervision when you are young.
Write a paragraph discussing the explainer. Start with a statement like "Growing up is like learning to ride a bike." Then explain the stages of learning to ride a bike.
Write a paragraph discussing the explained. Start with a statement that gives an overview of what the two share. In the example above, you might say something like "Growing up also involves getting greater and greater freedom as you become more confident." Then explain the steps of the explained in a way that parallels the explainer.
Discuss the differences. Sometimes there is a very important aspect of the explained that doesn't match up with the explainer. For example, in the above essay you eventually completely learn to ride a bike, but you never stop growing up and learning new things. You may want to draw attention to this important distinction.