When studying motion you want to know if the object is moving at a constant rate or if the rate is changing. Rate is a numerical measure of how fast you are going. You frequently also need to know the direction in which the object is moving. Different terms are used to describe these different types of motion.
For example, you are driving in your car at 30 m/s. If this is all the information given, you would be driving at a "speed" of 30 m/s. The rate is given but no direction. However, if you said you were driving 30 m/s to the east, you would now say your "velocity" is 30 m/s east. Velocity is considered a vector, which means it has both a magnitude (30 m/s) and a direction (east). The direction can also be indicated by an angle in degrees. For example, if you were going 20 m/s NE, you could also say you were going 20 m/s at an angle of 45 degrees.
When looking at acceleration, there is always a change in velocity over a certain period of time. So if you are in your car and you press on the gas pedal, you will start going faster. Conversely, if you press on the brake pedal, you will start slowing down. Acceleration is also a vector, so it is expressed as both a rate and a direction. With acceleration, the direction can be indicated either by a compass heading (N, S, E, W), by an angle, or by a positive or negative sign.
For example, you are driving East at a velocity of 22 m/s and increase your speed to 37 m/s in 5 seconds. Your acceleration would be: (37 m/s - 22m/s)/5 s = 3 m/s/s east or a positive 3 m/s/s or 3 m/s/s at 0 degrees.
Velocity is basically how fast an object is moving. It's not the same as speed because it isn't just how fast that object moves from one place to another but also considers direction. It has vector quantity. What this means is that velocity isn't just how fast or how slow an object is moving but in which direction it moves. An example would be that you were travelling 30 miles per hour due east.
Acceleration describes how velocity changes over a period of time. You take the velocity and divide it by how long it takes for the change in velocity to occur. So it's you travelling at 30 miles per hour due east and how this position changes over time.