(BW) Bandwidth is a measure of frequency range, the range between the highest and lowest frequency allowed in the signal. For analog signals, which can be mathematically viewed as a function of time, bandwidth is the width, measured in Hertz of a frequency range in which the signal's Fourier transform is nonzero.

A higher bandwidth is used for the reduction of chemical shift artifacts (lower bandwidth - more chemical shift - longer dwell time - but better signal to noise ratio). Narrow receive bandwidths accentuate this water fat shift by assigning a smaller number of frequencies across the MRI image. This effect is much more significant on higher field strengths. At 1.5 T, fat and water precess 220 Hz apart, which results in a higher shift than in Low Field MRI.
Lower bandwidth (measured in Hz) = higher water fat shift (measured in pixel shift).

See also Aliasing, Aliasing Artifact, Frequency Encoding, and Chemical Shift Artifact.

An image in which the signal from two spectral components (such as fat and water) is 180° out of phase and leads to destructive interference in a voxel.
Since fat precesses slower than water, based on their chemical shift, their signals will decay and precess in the transverse plane at different frequencies. When the phase of the TE becomes opposed (180°), their combined signal intensities subtract with each other in the same voxel, producing a signal void or dark band at the fat/water interface of the tissues being examined.
Opposed phase gradient echo imaging for the abdomen is a lipid-type tissue sensitive sequence particularly for the liver and adrenal glands, which puts a signal intensity around abnormal water-based tissues or lesions that are fatty. Due to the increased sensitivity of opposed phase, the tissue visualization increases the lesion-to-liver contrast and exhibits more signal intensity loss in tissues containing small amounts of lipids compared to a spin echo T1 with fat suppression. Using an opposed phase gradient echo also provides the ability to differentiate various pathologies in the brain, including lipids, methaemoglobin, protein, calcifications and melanin.

See also Out of Phase, and Dixon.

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.

(FID) A free induction decay curve is generated as excited nuclei relax. The amplitude of the FID signal becomes smaller over time as net magnetization returns to equilibrium. If transverse magnetization of the spins is produced, e.g. by a 90° pulse, a transient MR signal will result that will decay toward zero with a characteristic time constant T2 (or T2*); this decaying signal is the free induction decay.
The signal peaks of the echoes fall onto this T2 decay curve, while at each echo the signals arise and decay with T2*. The typical T2 relaxation times being of the order of 5-200 ms in the human body. The first part of the FID is not observable (named the 'receiver dead time') caused by residual effects of the powerful exciting radio frequency pulse on the electronics of the receiver.

(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.