You and the computer take turns popping balloons. On your turn, you can pop as many balloons as you like from any one row. When you've finished your turn, click the "Done" button on the left. The object of the game is to force your opponent to pop the last balloon.
If you want the computer to play first, click the "Done" button before you make your first move. The computer will remember your score for 7 days, but if you want to start again at level 1, click "Restart Game".
In certain situations, you can force a win. I'll call a position P a winning position if you can guarantee to win if you leave the game in position P after your move. For instance, (1 1 1) is a winning position, whereas (1 1) is not. Here, each number represents the number of balloons in a row.
The End Game
Generalising, if all the rows contain only one balloon and if there are an odd number of rows, then you have a winning position (for an even number, you have a losing position). On the other hand, if at the start of your move all but one row contains only one balloon, then you can win by reducing to the above situation of an odd number of rows containing only one balloon.
As long as n > 1, then (n n) is a winning position.
The strategy for more rows is not so simple to describe. I'll start you off by mentioning that (1 2n 2n+1) is always a winning position, and that any combination of winning positions is a winning position. For example, (1 4 5 6 6) is a combination of the winning positions (1 4 5) and (6 6).
If you want to read more about this game, and find out the perfect strategy, I recommend section 9.8 of An Introduction to the Theory of Numbers, by Hardy and Wright, 5th Edition, OUP, 1979. There is also a good Wikipedia article on Nim, a game equivalent to Balloon Blaster.