1. A form of narration: A story (usually a personal recount) that has a large build up, but doesn't go anywhere. Or, if it does go anywhere, it's nowhere worth going.
2. Used in a derogatory form (best delivered with extreme sarcasm and/or ) about a subject to express to someone that you don't care about what they are talking about.
Tom: "Oh- this awesome thing happened to me the other day, right- so I went to the shop, right."
All: "... riiiight?"
Tom: "Yeah, I was at the shop. And there was gum on some dude's shoe."
All: "That was such a Tom Story."