Anyway. Part 1: Peek, Peak, and Pique - with definitions (courtesy of Dictionary.com) and handy illustrations!
Peek: (V) to look or glance quickly or furtively, esp. through a small opening or from a concealed location; peep; peer.
Look, this dude is peeking through your window. What an asshole. But I bet he can use the right word for what he's doing.
Peak: (N) 1. the pointed top of a mountain or ridge.
2. a mountain with a pointed summit.
3. the pointed top of anything.
4. the highest or most important point or level: the peak of her political career.
The definitions continue but really it's more of the same. It's either the top, the pointy bit, or something that is considered both tall and pointy. The verb form refers to getting to the highest pointy bit.
Pique: (V) 1. to affect with sharp irritation and resentment, esp. by some wound to pride: She was greatly piqued when they refused her invitation.
2. to wound (the pride, vanity, etc.).
3. to excite (interest, curiosity, etc.): Her curiosity was piqued by the gossip.
4. to arouse an emotion or provoke to action: to pique someone to answer a challenge.
See, in the picture? I'm bored until he piques my interest by mentioning dragons (I love dragons!). He is not PEAKING my interest, nor is he PEEKING it.
The worst offense with these 3 homophones usually consists of people claiming they have 'peaked' someone's interest or curiosity. An argument could be made that they really mean peak in the verb sense, that they have raised my interest to its highest possible point, but context clues usually reveal that they merely mean they have aroused my curiosity. If they had, in fact, peaked anything of mine the response would be - shall we say - rather noticeable.
So as much as I want to give people the benefit of the doubt I know that 99% of the time they are simply getting confused, when writing, that these are actually 3 completely different words that happen to sound the same. Guys on dating sites will write something in their profile or in an email referencing their desire to peak or even peek me in some way but all they get is the lonely sound of me never, ever responding.