It's when you mistype "awesome" in text, IM's, or any other typing devices. But none of that matter, it's still Awesome as it is. Awesdome somehow sounds more awesome and cooler than awesome itself.
Text 1: "How was last night?"
Text 2: "Awesdome"
Text 1: "the fuck?"
Text 2: "I meant awesome!! you piece of shit; you slept with me!!!"
Text 1: "Wait, I thought that was... (looks at his a-hole) OH MY GOD!"