A new Korovkin type theorem is established, giving a system of conditions which insures the convergence in \(C[0,1]\) of a sequence \((L_{n})_{n \geq 1}\) of positive linear operators to its limit operator denoted by \(L_{\infty}.\) The approximation error \(| L_{n}(f,x)-L_{\infty}(f,x) |,\) \(f \in C[0,1],\) \(x \in [0,1],\) is estimated by the first order Ditzian-Totik modulus of smoothness. The converse result of the Korovkin type theorem is also given. As applications local and global quantitative estimates are obtained for \(q\)-Bernstein type operators which reproduce the functions \(e_{0}(x) = 1\) and \(e_{j}(x) = x^{j},\) \(x \in [0,1]\) and \(j \geq 1.\)