Hello A–C, welcome to UsingEnglish!
To "progress" something is to cause it to move on to its next natural stage; or merely to begin doing whatever will cause it to move on to its next natural stage.
The exact meaning would depend on the context. If you were progressing an application, for instance, you would be doing whatever you usually did to applications, and then making sure it went on to the next stage in the process.
(Since its meaning is conveniently vague, it's often used when we want to suggest that we've done more of a task or job than we really have.)