We discuss BPS states preserving 1/4 supersymmetries in N = 4 supersymmetric Yang-Mills theory as M2-branes holomorphically embedded and ending on M5-branes. We use techniques of electrodynamics to find the M2-brane configurations and give some explicit examples. In case that the IVI2-brane worldsheet has handles, the worldsheet moduli of the M2-brane is constrained in a discrete manner. Several aspects of multi-pronged strings in type IIB string theory are beautifully reproduced in the M-theory description. We also discuss the relation between the above construction of M2-brane configuration and the D2-brane dynamics in type IIA string theory.