A vector is a direction and magnitude (length). Vectors are used to represent many different things - light source directions, surface orientations, relative distances between objects, etc.
We normally describe a vector as a triplet of (X, Y, Z) values - (vx, vy, vz) represents a vector that points vx units in the direction of the X axis, vy units in the direction of the Y axis, and vz units in the direction of the Z axis.
e.g., (2, 0, 0) is a vector pointing in the direction of the X axis, 2 units long. (1, 1, 0) is a vector pointing at a 45 degree angle between the X and Y axes, 1.414 units long.
The magnitude of a vector (x,y,z) is its "Euclidean" length - the square root of x2 + y2 + z2.
Two vectors can be combined by adding their corresponding components
i.e. (vx, vy, vz) + (wx, wy, wz) is (vx+wx, vy+wy, vz+wz).
Or, written more expansively:
| vx | | wx | | vx + wx | | vy | + | wy | = | vy + wy | | vz | | wz | | vz + wz |
The result is a vector that is equivalent to sticking the vector W onto the end of vector V, and creating a new vector from the beginning of V to the end of W.
A vector can be multiplied by a single number (a "scalar") to change its length without changing its direction.
| vx | = | s * vx | s * | vy | = | s * vy | | vz | = | s * vz |