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.

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.

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.

(GRAPPA) GRAPPA is a parallel imaging technique to speed up MRI pulse sequences. The Fourier plane of the image is reconstructed from the frequency signals of each coil (reconstruction in the frequency domain).
Parallel imaging techniques like GRAPPA, auto-SMASH and VD-AUTO-SMASH are second and third generation algorithms using k-space undersampling. A model from a part of the center of k-space is acquired, to find the coefficients of the signals from each coil element, and to reconstruct the missing intermediary lines. The acquisition of these additional lines is a form of self-calibration, which lengthens the overall short scan time. The acquisition of these k-space lines provides mapping of the whole field as well as data for the image contrast.
Algorithms of the GRAPPA type work better than the SENSE type in heterogeneous body parts like thoracic or abdominal imaging, or in pulse sequences like echo planar imaging. This is caused by differences between the sensitivity map and the pulse sequence (e.g. artifacts) or an unreliable sensitivity map.

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

The Gibbs or ringing artifact appears as a series of lines in the MR image parallel to abrupt and intense changes in the object at this location. This artifact does not occur visibly on smooth objects. This artifact is caused by the Gibbs phenomenon, an overshoot or ringing of Fourier series occurring at discontinuities.
In the spinal cord, a small syrinx can be simulated by the Gibbs phenomenon. Gibbs artifacts are also seen in other regions, for example the brain//skull interface.
Fine lines visible in an image may be due to undersampling of the high spatial frequencies, respectively incomplete digitization of the echo.
With more encoding steps the Gibbs artifacts is less intense and narrower. Therefore, e.g. the artifact is more intense in the 256 point dimension of a 256x512 acquisition matrix.

This problem can only be resolved by smoothing filters (LanczosSigmaFactor, 2-D Exponential Filtering, Gegenbauer Reconstruction etc.) or with a higher acquisition matrix and/or a smaller FOV, to smooth the object.

See also Gibbs Phenomenon and Apodization.