Here's the official definition. For any vector v and any scalar c:
  • if c is positive, then cv is the vector in the same direction as v and with length c times the length of v
  • if c is negative, then cv is the vector in the direction opposite v and with length -c times the length of v
  • if c = 0, then cv is the zero vector: c0 = 0.

The diagram shows sample scalar multiples for a single vector v and scalars c = 2, c = -2 and c = 0.

To multiply a vector by a scalar, you basically "scale" that vector, i.e. you change its length by a factor the size of the scalar. It's a little more complicated than that, though: what happens if the scalar is negative?
If one vector is a scalar multiple of another vector, then it's parallel to that vector. The converse of that statement is also true: if two vectors are parallel, then one is a scalar multiple of the other.
In this diagram, the lighter vector is a scalar multiple of the darker one. You can adjust the scalar with the slider or change the darker vector by dragging its head.
If you have two parallel vectors and you know their lengths, how could you use those lengths to write one of the vectors as a scalar multiple of the other? What happens if one of the vectors is the zero vector?