Translation

A translation slides a shape across the plane without turning it or flipping it. Every point moves the same distance in the same direction. That direction-and-distance is captured by a column vector \begin{pmatrix} a \\ b \end{pmatrix}, read as "a across, b up".

In coordinates, the vector is just added to each point:

(x, y) \;\longrightarrow\; (x + a,\; y + b)

Because every point moves by exactly the same amount, the shape's size, angles and orientation are all unchanged — the image is an identical copy, just in a new place.

A translation is described by a column vector \begin{pmatrix} a \\ b \end{pmatrix}.
Each point moves by it: (x, y) \to (x + a,\; y + b).
The vector from a shape to its image is the same from every point.
To undo a translation, translate by the negative vector \begin{pmatrix} -a \\ -b \end{pmatrix}.

Seeing it on a grid

Take a triangle and translate it by \begin{pmatrix} 4 \\ 2 \end{pmatrix} — 4 across, 2 up. Step through the figure: the arrow shows the vector, and every vertex lands 4 right and 2 up from where it started. The image is the same triangle, just slid.