Nexus 2.1 and Newer
Users of Nexus 2.1+ can click on their username in the top right hand corner of the Nexus user interface. This will reveal a link to a Profile. Selecting Profile will show the Summary panel in the Profile tab. It contains a link that will bring up a dialog, which allows the user to change the account password.
Note that this only applies when using the built in XML based authentication of Nexus and not an external one like LDAP or Atlassian Crowd. All external authentication stores are read-only.
Nexus 2.0 and Older
In order to change your password as a Nexus user in releases prior to 2.1, press on the Change Password link in the Security menu in the left hand navigation panel.