The mess is a sort of club on a base, like there is usually an officer's mess and also a sergeant's mess or NCO's mess, and sometimes there are more than two based on rank. As for No. 1, that would be the best uniform, No. 2 not so smart, and in some services there may also be a No. 3 uniform which is even less formal. Ergo, No. 1 mess kit means the best uniform you would wear when eating a formal meal in the mess. What that actually means will depend on the service, regiment, etc., not to mention which country's forces we are discussing.

If you aren't going in military uniform it's a huge irrelevance. So, DJ or No. 1 mess kit essentially means DJ, or if you are American a tux is a DJ. I'm editting this to say I noticed that it also gave a doublet as a third alternative, and to say that I agree entirely with MacMillan's answer but thought you might be wondering what the actual terms meant.