Most people think this means getting faster, but it is more precisely defined.
When a car is getting faster, how long it takes to increase its speed (or velocity) is important. A car with good acceleration increases its velocity in a short time. Many cars are judged in this way.
e.g. The 2001 Honda Accord 3.0 V6 goes from 0-60 mph in 7.8 seconds, whereas the Lotus Elise 111 does 0-60 mph in 5.8 seconds.
Taking two seconds less means the Lotus has better acceleration.