If the receiving RF coil is sensitive to tissue signal arising from outside the desired FOV, this undesired signal may be incorrectly mapped to a location within the image, a phenomenon known as aliasing. This is 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 sampling frequency should be at least twice the frequency being sampled. The maximum measurable frequency is therefore equal to half the sampling frequency. This is the so-called Nyquist limit. When the frequency is higher than the Nyquist limit, aliasing occurs.
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. This signal will be mapped, or wrapped back into the image at incorrect locations, and is seen as artifact.

See also Aliasing Artifact.

(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.

A spectral line is a particular distinct frequency or narrow band of a frequency set. The resonance of this frequency occurs corresponding to a particular chemical shift. Theoretically, the frequency of a pure sine wave displays sharp spectral lines at the point of Larmor frequency. In reality, the spectral lines spread into a blurred peak, caused by field inhomogeneities and spin-spin effect.

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

A data truncation artifact may occur when the interface between high and low signal intensities is encountered in one imaging plane. The 2D-FT techniques transform the MR signal to spatial intensity image data with frequency and phase information encoding each axis in the plane of the scan. This artifact is found in both frequency and phase axes. Artifactual ripples adjacent to edges in an image or sharp features in a spectrum, caused by omission of higher frequency terms in Fourier transformation, particularly with the use of zero filling to replace unsampled higher frequencies.
Complex shapes are specified by series of sine and cosine waves of various frequencies, phase and amplitude. Some shapes are more difficult to encode than others. The most difficult shapes to represent with Fourier series of terms are waveforms with instantaneous transitions, tissue discontinuities or edges. The low-frequency components of the series describe the overall shape of the step function. Higher frequency components are needed to describe the corners if the step function more accurately. If not enough samples are taken, these areas cannot be accurately represented. The truncation of the infinite data series results in a ringing artifact because of the inability to accurately approximate this tissue discontinuity with a shorter truncated data set. Therefore, the ringing that occurs at all tissue boundaries on MR is called truncation artifact.

This problem can be easily resolved by taking more samples - a higher acquisition matrix and/or a smaller FOV. See Gibbs Artifact and Gibbs Phenomenon.

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.