How can acceleration be negative?

Acceleration is a vector...meaning that it has a magnitude (a "number"/size value assigned to it) as well as a direction (typically indicated by a + or - sign)."Negative" acceleration typically happens in two cases. One of them is from slowing down, and one of them is based on frames of references.

1) Slowing down
This "negative" acceleration is actually called deceleration. It means that over time, you are slowing down instead of speeding up- your velocity is continously decreasing. To show this decrease, mathematically we put a negative sign to indicate this.

2) Frame of reference
So let's say we assign a frame of reference of saying that going right is the POSITIVE + direction, and going left is the NEGATIVE - direction. Let's say you had a car going left...Well, you could either say that the car is accelerating to the left. Or, if you wanted to show this mathematically, you could put a negative sign in front of the value of acceleration to indicate that the car is going against the convention of right-is-positive.

You could also end up with a negative acceleration when trying to solve force problems...say you assumed the wrong direction in the free body diagram of your system. When you do the math, you will end up with a negative value. This indicates that your arrow is actually pointing in the wrong way. So getting a negative acceleration in these cases indicates some sort of frame of reference- that positive is going one way, and negative the other.

So, having a "negative" acceleration depends on the case you're dealing with, and it's not really something physical. It usually just results from conventional mathematical modelling.