An elastic nonwoven composite that contains an elastic film laminated to one or more nonwoven web materials is provided. The composite is formed by passing the film through a nip to bond the film to the nonwoven web material(s). Concurrent with bond formation, apertures are also formed in the elastic film. The apertures are of a size sufficient to provide a desired level of texture, softness, hand feel, and/or aesthetic appeal to the composite without having a significant adverse effect on its elastic properties. Aperture and bond formation are accomplished in the present invention by selectively controlling certain parameters of the lamination process, such as film content, bonding pattern, degree of film tension, bonding conditions, etc.