Each visibility sample is given a weight in the imaging step. This weighting can be used to account for differences in the density of sampling in different parts of the u-v plane, or to account for different noise variances in different samples, or to improve sensitivity to extended objects, etc. Here we briefly review some weighting schemes:
Natural weighting
This gives constant weights to all visibilities (or, more strictly, inversely proportional to the noise variance of a visibility). This weighting gives optimum point-source sensitivity in an image. However the synthesised beam-shape and sidelobe levels are usually poor.
Uniform weighting
This gives a weight inversely proportional to the sampling density function. This form of weighting minimises the sidelobe level. However the noise level can be a factor of 2 worse than natural weighting.
Super- and sub-uniform weighting
Uniform weighting computes the sampling density function on a grid that is the same size as the gridded u-v plane. This results in the synthesised beam sidelobes being minimised over the same field-of-view as the region being image. Surprisingly, making the field-of-view very large (bigger than the primary beam size) or very small (comparable to the synthesised beam) both cause uniform weighting to reduce to natural weighting. Super- and sub-uniform weighting decouple the weighting from the field size being imaged. Instead, the sidelobes in the synthesised image are minimised over some arbitrary field size, with this field being either smaller or larger than the field being imaged (for super- or sub-uniform weights respectively).
Robust weighting
Uniform weighting (including super- and sub-uniform weighting) minimising sidelobes, whereas natural weighting minimises the noise level. Robust weighting provides a compromise between the two, doing so in an optimal sense (similar to Wiener optimisation). See Dan Briggs' thesis for more information.
In signal processing theory, the optimum way to detect a signal of known form, which is buried in noise, is to convolve that signal with a ``matched filter''. This filter has an impulse response which is just the reverse of the form of the signal that is being detected. Applying this principle to detecting sources in radio interferometry, the optimum weighting for detecting a Gaussian source is to weight the visibility data by a Gaussian. This is often called `tapering'. Using a Gaussian weight will significantly increase the detectability of an extended source. However it also degrades the resolution. Gaussian weighting can be combined with any of the above weighting schemes to achieve some form of balance between sidelobes and sensitivity.

Miriad gives good (excessive?) control over the visibility weighting schemes, via three parameters and one option.

Miriad manager