Definition of bet sizing
Literally "size of bet". A bet sizing is the proportion of a bet to the size of a pot. For example, at flop, let's say pot is 10 euros. If you are playing online, it is likely that your poker room will have shortcut buttons for different "bet sizing". These buttons will be identified 1/2, 3/4 and pot. You may also have other options such as 1/4. These options are different "bet sizing".
Involvement of the bet sizing
The bet sizing to poker are an art form. Every time you bet, you will be studying your bet sizing. If your opponent is passive and you have a very strong hand, you will want to use a small bet sizing to get him to call your bet. On the other hand, depending on his level of thinking, a small bet might make it look like you are looking for a call with a very strong hand, which might prompt him to fold. Perhaps you'll recommend a very large bet (bet pot) to make him think you're bluffing him. And if you're up against a player who hates to be bluffed, maybe you'll go for an oversized bet size to maximize the money you've won with your hand.