It depends...
The simplest way is just to pickup a sheet of 3/4"
AC plywood from a home store and have them trim it to size for you. They might sell it by
mm thickness over there, so figure 20-22mm thick. It is study enough to support things over the edge of the dining table, has better sheer strength than
MDF and isnt too heavy to move around.
The
AC grade has one side that is smooth and the other less smooth, but it is dirt cheap ($25 or so here in the US for a 4x8 sheet from a store like Lowes). Many places will do simple cut, like trim it down to 4x6 for a small fee.
Next would be doing a skin and batten type board. For those, you use something like 1/4 or so plywood or Hardboard (sometimes called masonite...not
MDF though). Again, cut to size. The 1/4 inch thickness isnt strong enough on its own, so you will attach a frame of 1x2 stock to the backside (glued and screwed...or nailed I guess). This is lighter, but a bit more complicated to make.
Finally, you can make something that is modular. As opposed to having one big panel that you need to store, you make 3 or more small panels and connect them when in use. If you are careful, you can simply lay them next to each other, but I prefer to use a draw catch like these:
http://www.leevalley.com/US/Hardware/page.aspx?p=43997&cat=3,41399,41406&ap=1
Attach them to the edge and they pull the two panels together. They a strong enough to hold them tight during normal play, then when you are done, just open them up and you can more easily store the panels.
Some people like to use
MDF or particle board for the panels, but I dont. It is cheaper by a bit, but not as strong as plywood and a good bit heavier. In terms of specific types of plywood, most your hardwood plywoods are a bit excessive (birch, oak, maple...). Since we generally coat them in a paint to represent some sort of base board (blue to use for water, grey for asphalt, black for space...) and often add texture or a grass matt as well, regular Exterior grade
AC or cabinet grade
AB plywood is fine.
For the wood battens, Ill tend to use soft woods if it will be a single panel or an inexpensive hardwood (poplar or aspen here) if I am mounting those clasps to it for a bit of added pullout resistance on the screws.
You can get even fancier using hinges and what not, but if you are asking about wood...lets just keep it simple for now.