The multiplication of acquired MR data by a function smoothly tapering off at higher spatial frequencies so as to reduce ringing artifacts near edges in the corresponding image or spectrum due to truncation and Gibbs phenomenon. It is a form of filtering.

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.