Vector

Without a doubt, you are familiar with the concept of scalars. For example: 1, 2, $\sqrt{3}$, and $\pi$ are all examples of scalars. Scalars only have a magnitude; they don't have a direction.

A vector, on the other hand, has both direction and magnitude: displacement.

Notation

A common form to a show vector in is the column matrix:

$$\mathbf{v}=\begin{bmatrix} x \\ y \end{bmatrix}$$

This should be pretty self-explanatory: the $x$ value is the displacement along the $\mathbf{\hat{x}}$ axis, and $y$ is the displacement along the $\mathbf{\hat{y}}$ axis. See the graphic below for a live example.

To not confuse vectors with scalars, they need to be written differently. Arrow notation and boldface are the two most common methods to differentiate vectors from scalars, but are not the only methods.

$$\mathbf{v}=\vec{v}=\underset{\sim}{v}$$

This is very important, as vectors can also be 1D. Although the calculations for 1D vectors are the same as scalars, they represent completely different things conceptually.