Acceleration is a way to measure how fast something is speeding up. Suppose you are riding your bike. You start out going very slowly, hardly pedaling at all. Now you begin to pedal as hard as you can, to speed up - you are accelerating. Now that you are going at a normal speed, you stop pedaling so hard, and just pedal normally. You're still going, but you're not getting any faster, just going along at your normal speed. You're not accelerating anymore.
If you stop pedaling now, friction will work on your bike tires (and you'll have friction from the air, too), and you'll soon start to go slower. That's negative acceleration, or deceleration. You're still going, but you are slowing down.
One important cause of acceleration is gravity. Suppose you dive off a cliff or a high diving board. You will start off falling slowly, but as gravity pulls on you, you will speed up (accelerate) until you are going very fast.
The acceleration of Earth's gravity will speed you up at about 9.8 meters per second per second (9.8 m/s2, or 9.8 meters per second squared). That means that for every second you fall, you'll be going 9.8 meters/second faster.
Standing on the cliff before you jump, you're going zero meters/second. One second after you jump, you'll be going 9.8 meters/second. Two seconds after you jump, you'll be going 19.6 meters/second. Three seconds after you jump, you'll be going 29.4 meters/second.
Or check out this link to the Encyclopedia Britannica's article on acceleration.