See Aliasing Artifact.

(ASSET) ASSET is a parallel imaging technique of the SENSE type (image domain reconstruction).
Each coil element is sensitivity encoded and the covered spatial zone is mapped. By reducing the field of view in the phase encoding gradient direction the scan time decreases, but this images of each coil element contain foldover artifacts. The sensitivity profiles of the elements are used to calculate unfolded images.

See also Sensitivity Encoding, Generalized Autocalibrating Partially Parallel Acquisition.

(NPW / PNW - Phase No Wrap) If the receiving RF coil is sensitive to tissue signal arising from outside the desired FOV, this undesired signal may be incorrectly mapped, or wrapped back to a location within the image and is seen as artifact. This problem occurs in the phase encoding direction, where the phases of signal-bearing tissues outside of the FOV in the y-direction are a replication of the phases that are encoded within the FOV.
A user-selectable parameter maps this signal to its correct location outside the FOV, then discards any signal from outside the FOV before displaying the image. No phase wrap works by filling k-space to the same extent, using twice as many phase encoding steps. In order to be able to choose this parameter, in most cases more than an average is necessary.

See Foldover Suppression and Oversampling.

A problem occurs in the phase encoding direction, where the phases of signal-bearing tissues outside of the FOV in the y-direction are a replication of the phases that are encoded within the FOV. This signal will be mapped (wrapped, backfolded) back into the image at incorrect locations.
Foldover suppression (phase oversampling, no phase wrap) is a user-selectable parameter that maps this signal to its correct location outside the FOV, then discards any signal from outside the FOV before displaying the image. In order to be able to choose this parameter, in most cases more than an average is necessary.

See also Phase Wrapping Artifact and Oversampling.

Quick Overview
Please note that there are different common names for this artifact.

Backfolding always occurs due to wrong phase encoding caused by objects outside the planned FOV. Phase encoding gradients are scaled for the field of view only. Tissues outside the FOV do not get properly phase encoded relative to their actual position and 'wraps' into the opposite side of the image. The Backfolding artifact projects image contents which fall outside the imaging FOV back into the image; the back folded information thus reappearing on the other side of the image. In fact, information along the phase encoding direction can be viewed as projected onto a cylindrical screen with a circumference corresponding to the linear field of view dimension in the phase encoding direction.

See also Aliasing Artifact.