Of the two, scalars are the simpler quantity. A scalar is a measure of something that has only a magnitude to define it. A 10 meter length of cable is sufficient to define the length, whether we move it 2 meters to the left or 5 meters to the right. Also, 30 seconds is a scalar, and so is 25 degrees Celsius, and 30 liters. The magnitudes of these quantities are all that are needed to define them. Maybe to better understand that, we need to consider vectors as well...
Vectors are quantities that are defined by their magnitude AND their direction. This is the CRITICAL point to differentiate the two. Velocity is defined by the magnitude, but also requires the direction. If you are moving 10 m/s, you are moving 10 m/s IN SOME DIRECTION. This direction is required to fully define your velocity. The same is true when working with forces. You exert a magnitude of force in some direction, which fully defines the force. You can say that vectors are made up of a scalar component and a direction component.
When performing mathematical operations on scalars, you do so as you normally would. Run for 20 seconds, and then run for another 20 seconds, and you have run 40 seconds. Add 1 cup of water, and then add 2 cups, and you have added 3 cups. Since scalars are just composed of magnitude, you can simply operate on them as you always have. In fact, a lot of the math you have ever done, without problems, has been working with scalar quantities. As I said, just adding a fancy name doesn't make it any more difficult.
Working with vectors, however, is slightly more complex. You must always consider the direction component when doing the math. Many introductory physics courses will use examples of airplanes flying to demonstrate vectors. If the plane is flying 200 mph straight north (a vector), and the wind is blowing 50 mph straight east (another vector), you can imagine that the combined effect is that the plane is flying slightly at an angle rather than completely north. Actually, you can determine that the plane is actually travelling 206 mph in a direction of 14 degrees east of straight north, using simple trigonometry that you have already studied! (I will go over vector math in a future post.)
Hopefully, this post has more clearly explained the difference between scalar and vector quantities. Scalars are probably what you are most used to working with normally, whereas vectors require a little more thought, due to their direction component that always must be considered as well. Keep this key difference in mind as you work through your introductory physics homework problems.