A meromorphic function is the quotient of two holomorphic functions f/g. Technically, every holomorphic function is meromorphic, since you can always set g to 1.
Assuming g is nonzero, the zeros of g are isolated, (this will be proved in the next section), hence the poles of f/g are isolated. In a neighborhood without poles, f/g is once again differentiable, and holomorphic.
Remember that f and g have power series. Find the power series for f, and g, at a given point z0, and assume f has an mth order zero at z0, and g has an nth order 0. (Zeros and poles are described in the previous section.) The quotient has a laurent series, and is nonzero if m = n, or it has a zero of order m-n, or a pole of order n-m. An essential pole is not possible, hence E1/z is not meromorphic.