What does Japanese word Dai mean?

It could have lots of meanings. Most commonly its just a 'counter', you could say it means 'number' in English. 'Dai ichi', could mean '.... number one', where just about anything could go in the blank.

Another common usage of 'dai' is as an affix. It suggests 'largeness, intensity, etc'. 'Mondai' means problem, where 'daimondai' means a big problem. Of course different meanings mean different Words too, they are just pronounced the same, 'dai'.