Blackjack insurance is a side bet you can make when the dealer shows an ace. The dealer asks if you want to insure your hand against a dealer blackjack. The insurance wager pays 2:1, and is usually capped at half of your original wager.

The appeal is obvious. But in a standard six- or eight-deck shoe, insurance is usually a negative expected value bet, except in certain card-counting situations.
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That's because the unseen "ten" cards - tens, jacks, queens and kings - that could give the dealer a blackjack are just 31% odds. In a standard six-deck game, there are 311 unseen cards. Of those, 96 are ten-value, which means your insurance is going to win only about 30.9% of the time.
Do the math, and in a typical six-deck game, that gives the insurance bet a 7.40% house edge. The insurance loses 69.13% of the time. So no matter how you slice it, an insurance bet in a multi-deck game is a negative EV bet—meaning it loses more often than it wins.
The math works out the same when the dealer is showing an ace on a natural blackjack: in that case, insurance and taking even money are mathematically equivalent. Taking even money means you draw half your wager immediately, leaving the rest bet on the player blackjack. Your net result is exactly the same as making a half-wager insurance bet.
Now for the exception: if you know that the remaining shoe is ten-rich, insurance becomes favorable. Skilled card counters take advantage of this. One guide says that a six-deck player should trigger on a running count of 25 or higher, while another notes that eight-deck games carry a 7.5% house edge. But, if you are playing online, the shoe is shuffled after each hand, making insurance irrelevant.
So to sum up: for most cash-game blackjack, insurance is a bad bet. You can lower the house edge on your overall play by avoiding insurance. Only when you are playing in a casino, using perfect strategy, and the shoe is heavily weighted in your favor, does insurance become a positive expectation bet for some card counters.