A confirmation means your transaction has been included in a block on the blockchain. When it lands in a block, it has one confirmation. Each new block mined on top adds another.
Why count them?
Each block on top of yours adds more proof of work that an attacker would have to redo to remove your transaction. One confirmation already makes reversal difficult. Six make it extremely expensive and unlikely. The probability of a successful reversal drops exponentially with each block, as the Bitcoin white paper shows with a simple calculation.
Rules of thumb
0 confirmations: the transaction is still in the mempool. Fine for trusted friends or a cup of coffee, but it could still be replaced. Use the Lightning Network for truly instant payments.
1–3 confirmations: fine for most everyday amounts.
6 confirmations (about an hour): the traditional standard for large payments. Many exchanges use a similar number for deposits.
100 confirmations: required before a miner can spend newly created coins from a block reward.
How long does it take?
Blocks arrive about every ten minutes on average, but it is random. Sometimes two arrive within a minute, and sometimes an hour passes. If the network is busy, your transaction may wait several blocks for its first confirmation unless you pay a higher fee.
Reorganizations
Occasionally two miners find blocks at the same time, and one branch is abandoned. Transactions in the losing block go back to the mempool and are usually confirmed again quickly. This is called a reorg. One-block reorgs happen now and then. Deep reorgs are extremely rare, which is exactly why waiting for more confirmations gives you more certainty.