Peek, peak, and pique: they sound the same but mean very different things.
The first one we learn is peek: it has to do with looking, especially furtively or quickly or through a small space, as in "open the box and peek inside." It's both a noun and a verb; when you peek, you take a peek. Our advice for remembering this one is to keep in mind that you peek in order to see.
Peak is the verb you use to talk about reaching a maximum, or coming to a highest point, literally or figuratively, as in "The meteor shower will last for several days but will peak on Sunday." Its noun counterpart, which refers to various pointed or projecting parts, is more common: something that peaks reaches a peak. Just as every mountain has a peak, thinking of the peak—the highest point—is the way to remember that peak is the choice for reaching the highest levels. Associating the "a" in peak with the "a" in maximum or with a capital "A" (the most mountain-like of letters) can be helpful.
Pique is the oddball of this trio. We know the "ique" spelling from the likes of technique, antique, and unique, but pique nonetheless looks a little exotic. It comes from a French word meaning literally "to prick," but its earliest English use was as a noun. The noun is still used: a pique is a transient feeling of wounded vanity—a kind of resentment. As a verb, pique was (and still is, especially in British English) used to mean "to arouse anger or resentment in," as in "Their rudeness piqued me." Now, however, it's most often our interest or curiosity that gets piqued—that is to say, our interest or curiosity is aroused, as in "The large key hanging next on the wall piqued my curiosity."
Pique has another meaning too, though it's less common than any of those already mentioned. Pique sometimes is used to mean "to take pride in (oneself)," as in "She piques herself on her editing skills."
Master this trio, and you can pique yourself on your word skills.