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It depends. For instance, if the shortstop boots a ground ball and allows the batter to reach 1st base, it is an error, no base hit, and the batter is officially 0-1 in that at bat.
Now, let's say the batter hits the ball through the hole between 3rd and short, and while attempting to pick it up, the left fielder bobbles it, allowing the batter to reach 2nd base. Then you've got a single, with the batter reaching 2nd on the error by the left fielder. In this case, the batter is 1-1 in that at bat.
The only times a batter hits the ball in play and is not credited with an at bat are when he is credited with a sacrifice or a sacrifice fly.
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