This term originated through the use of scat, or nonsensical language. It is meant to be pronounced in a very loud and high pitch tone of voice with particular emphasis on the "wee" part of the word when one articulates it. It also works best when an individual is enduring a barrage of stress in that particular moment. Therefore, the word, when used correctly, relieves stress.
Alex: "Hey Kristy, how're you doing today?"
Kristy: "Not too well. I hate school and work! I just want to scream!"
Alex: "Have you tried the Aweeka method?"
Kristy: "Tried it?! I live by it! AwEEEEEEEEEEEEEEEEEEEEEEEEEka!"
(n.) When something is so unbelievably cute that it leads you to say "awe." A play on the idea that orgasms are an extreme state of pleasure, "awegasm" represents a moment when one is overcome by an extreme level of cuteness.
(adj.) Awegasmic.
When I stopped at the pet store today I had an awegasm while I was playing with the kitties.
Upon seeing my girlfriend's new facebook default, I had an awegasm. :3