On a gene expression array, where two samples are compared, the terms are used much like Jason described. There is usually a test sample (i.e. mutant) and a reference sample (i.e. wild type). It's fairly common to then construct ratios of expression with the test sample in the numerator, and the reference sample in the denominator. If these ratios are examined in log2 format, then positive ratios indicate enrichment of a transcript in the numerator, whereas negative ratios indicate enrichment in the denominator (often referred to as up-regulated or down-regulated, respectively). The terms "up-regulated" and "down-regulated" are somewhat unfortunate as they refer to a mechanism relative to a particular sample. The better way to think about it is enrichment. When comparing transcript concentrations in two samples on an array, all one can really say is that the transcript is enriched in one relative to the other. Whether up-regulation or down regulation is going on, is a hypothesis for further exploration in subsequent experiments.
Usually up-regulated means more highly expressed compared to the reference where as down-regulated means expressed lower compared to the reference. But could you be more specific as to what you are referring to i.e. expression array, proteomics, RNA-seq and what is being looked at genes, RNA, etc?