Exploding dice is a game mechanic most often found in tabletop role-playing games. It is comparatively uncommon, and doesn't appear in any games that most people have heard of. However, it is quite satisfying when it kicks off, and adds an occasional exciting bonus to dice rolls.
Exploding dice are a rule under which, when a player rolls the maximum value on a die, they roll again and add the values; if they roll max a second time, repeat, and so on. If you are rolling comparatively small dice, such as a D4, this can be a fairly common occurrence. The more sides a die has, the less likely it is to explode; most games using dice larger than a D6 don't have a big reason to bother with exploding dice, although World of Darkness and Chronicles of Darkness series (e.g., Vampire: The Masquerade) do use exploding dice with a D10 system.
When rolling multiple dice under an exploding dice scenario, once usually tracks all the dice separately -- roll the dice, and re-roll any dice that hit their max value, being careful to keep track which dice have been rolled how many times. However, this may vary based on the game you are playing.