For a two-dimensional surface M2 in the four-dimensional Euclidean space E4 we introduce an invariant linear map of Weingarten type in the tangent space of the surface, which generates two invariants k and κ. ¶ The condition k = κ = 0 characterizes the surfaces consisting of flat points. The minimal surfaces are characterized by the equality κ2 - k = 0. The class of the surfaces with flat normal connection is characterized by the condition κ = 0. For the surfaces of general type we obtain a geometrically determined orthonormal frame field at each point and derive Frenet-type derivative formulas. ¶ We apply our theory to the class of the rotational surfaces in E4, which prove to be surfaces with flat normal connection, and describe the rotational surfaces with constant invariants.