A vector is formally defined as an element of a vector space. In the commonly encountered vector space $\mathbb R^n$ (i.e., Euclidean $n$-space), a vector is given by $n$ coordinates and can be specified as $(x_1,x_2,\ldots,x_n)$. Vectors are sometimes referred to by the number of coordinates they have, so a $two$-dimensional vector $(x_1,x_2)$ is often called a two-vector, an $n$-dimensional vector is often called an $n$-vector, and so on.
Vectors can be added together (vector addition), subtracted (vector subtraction) and multiplied by scalars (scalar multiplication). Vector multiplication is not uniquely defined, but a number of different types of products, such as the dot product and cross product can be defined for pairs of vectors.
The following simulation shows different properties and basic operations with vectors in two dimensions.
Things to try:
The University of Queensland