The acceleration of electrons by a Laguerre-Gaussian(LG) beam in vacuum is studied. It is shown that only the longitudinal electric field of the LG beam with mode indices p and l=1 can be used to accelerate electrons. The linearly- and circularly-polarized LG beams with mode indices p and l=1 effectively play the role in laser electron acceleration. Some physical characteristics, such as phase and group velocities of the axial optical field, and energy gain of electrons etc., are discussed. The analytical expressions for the phase and group velocities of the axial optical field, accelerating potential and energy gain etc. are derived and are used to make numerical analysis.