I have used the same exact first day activity every year so far. During my new teacher training we learned that we shouldn't start with rules and stuff on the first day. Now I don't think that this is necessarily the "right" thing to do, but it is something that stuck with me.
I remember stressing out SO much over the first day that first year. I wanted to find the perfect thing to do and I didn't really know what kids would like or whether they would think whatever I came up with was lame or boring. In hindsight, I stressed out over it way too much. I finally came across this game the night before school started. It involves playing cards so the morning of my very first day I had to try to hunt down as many sets of cards I could find (which happened to be miniature Snoopy, Mickey & Winnie the Pooh cards that I still use).
When the kids walk in the desks are all in tables and they can sit wherever they want. If necessary I will even out the tables a bit before we begin. The name of the game is 31-derful. I think that this might actually be a solitaire type game, but I had never heard of it.
In my honors class whichever group finishes first earns a HW/extra credit pass for each person (in these classes usually everyone finishes so I do the first group to add some competition). In my regular classes I will usually give passes out to any group that finishes.
I like this for the first day because it sets a very low key vibe, which is completely me. They can come in and sit wherever they want, I'm not lecturing for the period, and they get to chat with friends a little bit as they are engaged in some math work that definitely requires thinking. As they work I walk around and float in and out of groups and chat a little about what is/isn't working. Sometimes they get totally off track and we laugh as they decide to just start all over. Giving up isn't an option. Hard work is rewarding. All of this actually sums up my class in general pretty well.
The other part, which I'm willing to admit, is that this makes for an easy first day for me which I need. There's enough going on the first day as is and I just don't want to have to worry about making copies and talking about my class for five periods which is exhausting (I hate rule/expectation/syllabus day). By doing this, all I need to worry about is explaining the activity and getting my cards off the shelf. Simple.
So does anyone have a first day activity that you think sums up what your class is like?