Definition of chip count
A chip count is when a player asks the dealer or tournament director for the exact value of an opponent's stack (how many chips does he have?). This only happens when playing live (online, you can quickly know the size of your opponent's stack at any time). The chip count usually occurs late in tournaments when the situation becomes difficult.
Why do we ask for a chip count?
Usually, the player requesting a chip count has a difficult decision to make. If a player is all in before me and I have the aces, I don't need to know how big his all in is, I call the bet! But let's imagine the following situation. You are at the final table of a tournament. You are the chip leader on the table. You're very active and you're doing a lot of 3bets and raises. A player opens in an early position, you're on the button with QA and you go for a 3bet. The player says, "I'm all in!". Here you're going to call a chip count. You're a chip leader and your stack is valuable, but if what he adds in chips after your raise is very little, you'll have no choice but to call the all in. You need to know the size of his stack to make the right decision.