Usually this is in reference to WordPress, Drupal or other software installation.
Updates are within a software version. For example we update WordPress from 3.0 to 3.1. What’s important is the first number remains the same. Updates are simpler tasks and are often done to patch security vulnerabilities.
Upgrades are more complex and often require more attention and testing. For example from Drupal 6.x to Drupal 7.x. What’s important is that the first number changes. Usually there are more major revisions between one version and the next and there may be a waiting period to upgrade if certain plugins are required. Upgrades tend to extend the capabilities of the underlying software, add features, and or operate faster.