Insurance
When the dealer shows Ace, there is always the worry that there is a Ten, Jack, Queen, or King facing down. Casinos know players worry, so they offer "Insurance" to players in case of a dealer blackjack – which is essentially just a side-bet on the dealer’s down-card. To buy insurance, the player may place a wager in an amount up to half of the original wager. If the dealer has blackjack, this bet is paid out 2:1. If the dealer does not have blackjack, the player loses the bet.

Example:
Suppose a player bets $100 and is dealt [9, 6] and the dealer is dealt [A, ?]. The dealer will immediately offer the player insurance. Now suppose the player buys insurance for the maximum amount of $50 (half the original bet).

Scenario A: Dealer has blackjack. The player wins $100 on the insurance bet, but loses $100 with his [9, 6] to the dealer’s blackjack.
Scenario B: Dealer does not have blackjack. The player loses the $50 insurance bet, and the hand is played out.

Even Money
Even Money is essentially a special case of insurance. Even Money is offered to a plyer with blackjack when the dealer has an Ace showing. Taking Even Money means that the player agrees to accept a guaranteed 1:1 payout on blackjack instead of either (a) a push if the dealer has blackjack or (b) a 3:2 payout if the dealer does not have blackjack. As it turns out, when the player has blackjack “taking Even Money” is equivalent to “buying Insurance” for the maximum amount.

Example:
Suppose the player bets $100 and is dealt blackjack. The dealer is dealt [A, ?].
Even Money: The player is paid $100 and the hand is over, regardless of the dealer hand.
Insurance: The player wagers $50.
Scenario A – If the dealer has blackjack, the player receives $100 on the Insurance bet, and the hand is a push. Net effect +$100.
Scenario B – If the dealer does not have blackjack, the player loses $50 on the Insurance bet, but wins $150 (3:2) with blackjack. Net effect +$100.

Because insurance and even money are identical when the player has blackjack, in practice only Even Money is offered to a player with a blackjack when the dealer shows an Ace. Insurance is offered to everyone else.

A Bad Bet
Taking Insurance or Even Money is a bad bet. Why? First, let’s look at the payout structure:

Win: Receive 2x original bet
Lose: Lose original bet Now, let’s look at the likelihood of winning or losing:

Probability of Winning: The probability of winning is the probability that the dealer has a Ten, Jack, Queen, or King facing down. This probability can be approximated by counting the number of those cards in shoe and dividing by the number of cards in shoe. Each deck has four of each of those cards for a total of 16. And each deck has 52 cards. So regardless of the number of decks in the shoe, the probability of winning can be approximated by 16/52 or 4/13, less than 31%.
Probability of Losing: The probability of losing is just 1 – 31% = 69%.

So if you win 2 units 31% of the time, but lose 1 unit 69% of the time, then in the long-run you will lose:

31% * 2 + 69% * -1 = -7%

So for every $100 of insurance, you can expect to lose about $7.

Example:
Let’s consider the situation in which you buy $1 of insurance 100 times. You can expect to win 31 times, and you will receive $62. You can expect to lose 69 times, and you will lose $69. So in the end, you will have bought $100 worth of insurance, and you can expect to lose $69 – $62 = $7.