I think for me, it's my connection to my Dad. He died when I was 16 (I'm 51 now), and my memories of him are closely tied to the Bills. As a little guy, I'd be out playing and come running in the house to ask what the score was and whether we were wearing white or dark (it was black and white TV in those days).
I started following them on my own in 1968, and going to games with him in War Memorial are very fond memories. In 1970, he sprang for the most expensive tickets, 3 of them at $7 each under the roof. I looked at them every day pinned on the board in the kitchen. When game day came, my Dad took me for a walk and said he couldn't go because he was having chest pains. I was so disappointed and ended up having to go with my mother and my sister. It was horrible....drunks spilled wine on me, burned cigarette holes in our blanket, and OJ got his knee torn up on a kickoff return and was lost for the season. The Bills got trashed by the Bengals.
But it never made me lose faith in the team, because it was a big connection to my Dad. When I lost him, I kept following the Bills, even after moving to Syracuse and having to listen to the games on a staticy radio feed of the games from Rochester. Bills games usually weren't televised in Syracuse back in the mid-80s, and that's probably a good thing! But I kept following them, and ended up going to the last Super Bowl to see them lose yet again. But no matter, I'll always be a Bills fan and always think of my Dad with every game!