In this article a generalized finite difference method (GFDM), which is a meshless method based on Taylor series expansions and weighted moving least squares, is proposed to solve the elliptic interface problem. This method turns the original elliptic interface problem to be two coupled elliptic non-interface subproblems. The solutions are found by solving coupled elliptic subproblems with sparse coefficient matrix, which significantly improves the efficiency for the interface problem, especially for the complex geometric interface. Furthermore, based on the key idea of GFDM which can approximate the derivatives of unknown variables by linear summation of nearby nodal values, we further develop the GFDM to deal with the elliptic problem with the jump interface condition which is related to the derivative of solution on the interface boundary. Four numerical examples are provided to illustrate the features of the proposed method, including the acceptable accuracy and the efficiency.