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.

A wrap around artifact (also called backfolding artifact or aliasing artifact) is produced by inadequate sampling or digitization. Wrap around artifacts result from digitizing fewer than two samples per period in a periodic function. Aliasing can occur in MR imaging whenever the scanned area extends beyond the field of view. These areas extending beyond the field of view boundaries are aliased back into the image to appear at artifactual locations.

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

Aliasing is an artifact that occurs in MR images when the scanned body part is larger than field of view (FOV). As a consequence of the acquired k-space frequencies not being sampled densely enough, whereby portions of the object outside of the desired FOV get mapped to an incorrect location inside the FOV. The cyclical property of the Fourier transform fills the missing data of the right side with data from behind the FOV of the left side and vice versa. This is caused by a too small number of samples acquired in, e.g. the frequency encoding direction, therefore the spectrums will overlap, resulting in a replication of the object in the x direction.
Aliasing in the frequency direction can be eliminated by twice as fast sampling of the signal or by applying frequency specific filters to the received signal.
A similar 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. Phase encoding gradients are scaled for the field of view only, therefore tissues outside the FOV do not get properly phase encoded relative to their actual position and 'wraps' into the opposite side of the image.

Use a larger FOV, RFOV or 3D Volume, apply presaturation pulses to the undesired tissue, adjust the position of the FOV, or select a small coil which will only receive signal from objects inside or near the coil. The number of phase encoding steps must be increased in phase direction, unfortunately resulting in longer scan times.
When this is not possible it can be corrected by oversampling the data. Aliasing is eliminated by Oversampling in frequency direction. No Phase Wrap (Foldover Suppression) options typically correct the phase encoding by doubling the field of view, doubling the number of phase encodes (to keep resolution constant) and halving the number of averages (to keep scan time constant) then discarding the additional data and processing the image within the desired field of view (but this is more time consuming).
Tissue outside this doubled area can be folded nevertheless into the image as phase wrap. In this case combine more than 2 number of excitations / number of signal averages with foldover suppression.
See also Aliasing, Foldover Suppression, Oversampling, and Artifact Reduction - Aliasing.

Undersampling is the decrease in data to increase image acquisition speed (shorter scan times without loss of quality - increased productivity - reduced cost of equipment). There are different strategies to decrease data without losing quality (e.g. reduction of the FOV in one or more spatial directions - RFOV).
Reduction in data normally is associated with an increase in aliasing (degradation of the SNR through backfolding of the entire noise spectrum), or with other artifact caused by missing data, which results in fine lines.