In parallel MR imaging, a reduced data set in the phase encoding direction(s) of k-space is acquired to shorten acquisition time, combining the signal of several coil arrays. The spatial information related to the phased array coil elements is utilized for reducing the amount of conventional Fourier encoding.
First, low-resolution, fully Fourier-encoded reference images are required for sensitivity assessment. Parallel imaging reconstruction in the Cartesian case is efficiently performed by creating one aliased image for each array element using discrete Fourier transformation. The next step then is to create an full FOV image from the set of intermediate images. Parallel reconstruction techniques can be used to improve the image quality with increased signal to noise ratio, spatial resolution, reduced artifacts, and the temporal resolution in dynamic MRI scans.
Parallel imaging algorithms can be divided into 2 main groups:
Image reconstruction produced by each coil (reconstruction in the image domain, after Fourier transform): SENSE (Sensitivity Encoding), PILS (Partially Parallel Imaging with Localized Sensitivity), ASSET.
Reconstruction of the Fourier plane of images from the frequency signals of each coil (reconstruction in the frequency domain, before Fourier transform): GRAPPA.
Additional techniques include SMASH, SPEEDER™, IPAT (Integrated Parallel Acquisition Techniques - derived of GRAPPA a k-space based technique) and mSENSE (an image based enhanced version of SENSE).

Partial averaging is a scan time reduction method that takes advantage of the complex conjugate of the k-space. The number of phase encoding steps of the acquisition matrix are reduced in the phase encoding direction.
Since negative values of phase encoded measurements are identical to corresponding positive values, only a little over half (more than 62.5%) of a scan actually needs to be acquired to replicate an entire scan. This results in a reduction in scan time at the expense of signal to noise ratio. The time reduction can be nearly a factor of two, but full resolution is maintained.
Partial Fourier averaging can be used when scan times are long, the signal to noise ratio is not critical and where full spatial resolution is required. Partial averaging is particularly appropriate for scans with a large field of view and relatively thick slices; and in 3D scans with many slices. In some fast scanning techniques the use of partial averaging enables a shorter TE thus improving contrast.
Partial averaging is also called Fractional NEX, Half Scan, Half Fourier, Phase Conjugate Symmetry, Single Side Encoding.

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

This artifact is caused by movements of the patient or organic processes taking place in the body of the patient. The artifact appears as bright noise, repeating densities or ghosting in the phase encoding direction.

There are different solutions for reduction of phase encoded motion artifacts.
Cardiac and respiratory gating, breath holding, sedation of the patient, presaturation pulses for flow artifacts (e.g. arterial pulsation, breathing), fast imaging sequences, etc.

See also Motion Artifact, Ghosting Artifact, Motion Compensation Pulse Sequences and Artifact Reduction - Motion.

The data acquisition beyond the FOV in phase encoding direction, with doubling the number of acquisitions and the scan time.

See also Oversampling and Aliasing Artifact.

Parts of the body, which extend beyond the Field of view in the phase encoding direction may still be visible in the image as artifacts.
Please note that aliasing in the frequency direction is also possible but can be eliminated in the same manner or by applying frequency specific filters to the received signal.

See Aliasing Artifact.