For example:
$$\rm{X=\sigma_x=NOT=|0\rangle\langle 1|+|1\rangle\langle 0|=\begin{bmatrix}0 & 1 \\ 1 & 0\end{bmatrix}}$$
$$\rm{Z=\sigma_Z=signflip=|0\rangle\langle 0|-|1\rangle\langle 1|=\begin{bmatrix}1 & 0 \\ 0 & -1\end{bmatrix}}$$
I do no understand how the matrix can be expressed in bra-ket and the way around.
Recall that kets $|\cdot\rangle$ represent column vectors; a bra $\langle\cdot|$ is a ket's row vector counterpart. For any ket $|\psi\rangle$, the corresponding bra is its adjoint (conjugate transpose): $\langle\psi| = |\psi\rangle^\dagger$. (For a refresher on this, see this question).
Kets and bras give us a neat way to express inner and outer products. The outer product of two vectors of the same size produces a square matrix. We can use a linear combination of several outer products of simple vectors (such as basis vectors) to express any square matrix. For example, the $X$ gate can be expressed as follows:
$$X = |0\rangle\langle1| + |1\rangle\langle0| = \begin{bmatrix} 1 \\ 0 \end{bmatrix}\begin{bmatrix} 0 & 1 \end{bmatrix} + \begin{bmatrix} 0 \\ 1 \end{bmatrix}\begin{bmatrix} 1 & 0 \end{bmatrix} = \begin{bmatrix} 0 & 1 \\ 0 & 0 \end{bmatrix} + \begin{bmatrix} 0 & 0 \\ 1 & 0 \end{bmatrix} = \begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix}$$
