A textured laminate structure and absorbent articles having a component formed of the textured laminate structure are disclosed. The textured laminate structure may have a first layer formed of a nonwoven web having a preexisting pattern of first apertures therethrough, a second layer, and one or more elastic members disposed between the first and second layers in a prestrained condition. The laminate structure may be discontinuously joined or bonded such that the first layer is joined or bonded to the structure in first portions and unjoined or unbonded to the structure in second portions. The structure provides a stretch laminate with tactilely and visually interesting features, and may be used to form components of a disposable absorbent article such as a side panel thereof.