The Hermitian adjoint of a matrix is the same as its transpose except that along with switching row and column elements you also complex conjugate all the elements. If all the elements of a matrix are real, its Hermitian adjoint and transpose are the same. In terms of components, $$\left(A_{ij}\right)^\dagger=A_{ji}^*.$$

For example, if $$A = \left(\begin{array}{c} 1\\ i\\ -2i\\ \end{array} \right)$$ then $$A^\dagger = \left(\begin{array}{ccc} 1&-i&2i\\ \end{array} \right).$$

A matrix is called Hermitian if it is equal to its adjoint, $A = A^\dagger$.

Try it for yourself by computing $B^\dagger$ if $$B=\left(\begin{array}{cc} 1&i\\ -5i&i\\ \end{array} \right).$$