But what is a '$4/$8 limit' game? Poker is generally played with three different betting forms. These are known as fixed-limit (or just limit for short), no-limit and pot-limit.
Let's start with limit, which is the most common form of poker for home games and cardroom cash games throughout the US. In a limit game you are only allowed to bet (or raise) the amount specified by the limit each time the action comes to you. Taking our example of a $4/$8 Hold'em game, before the flop you can either bet $4 or raise another $4 to $8 total. The next player then has the option of matching your bet or raising a further $4 etc. There is normally a restriction on the number of bets allowed in any one round (typically one bet and three raises or one bet and four raises).
On the flop the betting is the same. The first player may check or bet $4 and again the next player has the option of matching that bet or betting an additional $4 etc. However, for the turn and river the betting limits are doubled to $8 (hence $4/$8 game). Now the first player must either check or bet $8, and all subsequent bets will be in increments of $8.