A square matrix $$A$$ is diagonalisable if there is an invertible matrix $$P$$ such that $$P^{-1}AP$$ is a diagonal matrix. One can view $$P$$ as a change of basis matrix so that, if $$A$$ is viewed as the standard matrix of a linear map $$T$$ from a vector space to itself in some basis, it is equivalent to say there exists an ordered basis such that the standard matrix of $$T$$ is diagonal.
Not every matrix is diagonalisable over $$\mathbb{R}$$ (i.e. only allowing real matrices $$P$$). For example, $$\left[\begin{array}\ 0 & -1\\ 1 & 0\end{array}\right]$$ is not diagonalisable over $$\mathbb{R}$$.
Diagonalization can be used to compute the powers of a matrix $$A$$ efficiently, provided the matrix is diagonalizable.