What does it mean to introduce local coordinates near a point $p$? Let $p$ be a periodic point of period $m$ for a $C^1$ map $f$. Suppose that the differential $Df^m_p$ does not have 1 as an eigenvalue. Can someone explain me what does it mean the folowing facts:

Let us introduce local coordinates near $p$ with $p$ as the origin. In this coordinates $Df^m_0$ becomes a matrix.

What does it mean to introduce local coordinates near $p$? Why $Df^m_0$ becomes a matrix?
P.S I'm a beginner in this area, so please be patient with me :D Thank you!
 A: On some kind of spaces (topological manifolds, differentiable manifolds,...) we assume that around each point $p$ there is an open neighbourhood $U(p)$ which is homeomorphic (or diffeomorphic) to an open neighbourhood $V(p)$ of $\mathbb{R}^n$ (or $\mathbb{C}^n$, etc.). We thus say that our space is locally homeomorphic (diffeomorphic) to $\mathbb{R}^n$ (or $\mathbb{C}^n$, etc.). On $\mathbb{R}^n$ you can choose different types of coordinates (Cartesian, polar, cylindrical, spherical, etc.), which means that if you have a point in $\mathbb{R}^n$ you can attach $n$ numbers $x^1,\dots,x^n$ to identify it in the space (that is why you say that a point in the plane has Cartesian coordinates $(0,1)$ or whatever). Now if you call $\phi \colon U(p) \rightarrow V(p)$ the homeomorphism (diffeomorphism) above, then 
$$(x^1 \circ \phi)(p), (x^2 \circ \phi)(p), \dots, (x^n \circ \phi)(p)$$
are the coordinates of $p$. You can of course do the same for any point of $U(p)$ getting coordinates of points in $U(p)$, i.e. around $p$. $(x^i_{|p})_{i=1,\dots,n}$ is then called a local coordinate system.
