Datasets where different pieces of data can have different "weights", i.e. different importance.

When weights are put on cases/observations/rows, they may indicate different probabilities of selection or response in complex surveys; different accuracy of the underlying observations. A useful resource describing the various type of the observation weights is http://www.ats.ucla.edu/stat/stata/faq/weights.htm.

Another situation when the weights are attached to a unit (row in the data) arises in importance sampling when the weight is the ratio of the target density (that is difficult to sample from) to the convenient sampling density.

When weights are put on variables/columns/traits (regression weights, factor analysis weights, neuron input weights, etc.), these are model coefficients. Please tag your question with the model-specific tag in that situation, and avoid this tag when your weights are attached to variables/columns rather than observations/cases/rows.