Ultimately, no analogy will help us comprehend this doctrine. It transcends reason.
(By the way, transcending reason is not the same thing as being contrary to reason. What is contrary to reason falls within the jurisdiction of reason and is condemned by reason. What transcends reason is outside the jurisdiction of reason altogether.)
Now, I think some analogies are helpful. Augustine, for example, develops a good psychological analogy for the Trinity. But the analogies you've probably heard the most of are not very good. One of the most common analogies is water: Water is one thing, but it takes three forms: liquid water, water vapor, and ice.
Another common analogy is that of a person with several different roles: For example, a woman who is a mother, a daughter, and a wife all at the same time.
The Trinity is simple. God is like this pizza. There's a crust, cheese, and tomatoes. Now listen, this is gonna blow your mind: It's ALL THE SAME PIZZA! See, God is just like that.
Unfortunately, these analogies don't portray orthodox Trinitarianism so much as the heresy of modalism
Modalism, also called Sabellianism after the guy who promoted this error in the early church, is the view that there is one God who reveals himself to human beings in three different roles, or three different modes: the mode of Father, the mode of Son, and the mode of Holy Spirit.
Since these analogies show how one thing can also be three things, I think they are not completely useless. The problem is that they don't show how one thing can also be three persons. The analogy of the water shows how the same substance, water, takes three forms, or appears in three modes. The analogy of the person with three roles shows how one person can have three roles; this is a perfect illustration for modalism, but not for orthodox Trinitarianism.
It's understandable that people would want to affirm modalism. It is easy to make sense of it--a little too easy. It reduces the mystery of the orthodox doctrine to a comfortable, easily understood concept.
Of course, there's nothing wrong with easily understood views. Most of the best views, I tend to think, are quite easily understood if we only read the right books (e.g., C. S. Lewis. The problem with this view is simply that it isn't the correct view of the Trinity. It recognizes one God, but that's not the whole doctrine. It doesn't recognize three persons; it actually denies them.
For more on this, try listening to about 40 minutes of William E. Bell. The relevant lecture is available here. (But I cannot confirm that Bell is legally available here; I don't know who owns the copyright to his lectures, or whether that person gave permission for them to be on this particular website.)