To put it simply, a common application is an application or portion of an application that you can use on more than one of the programs that you create.
For example: a community organization wants to create five different programs to serve the needs of their community members. They decide to create a program for students who are graduating early, one for young women who pursue a math or science degree, students who plan to go into the arts, returning students, and Native American students. Because these different programs are geared towards different groups, they will each have portions of the application that are different. However, since all of the funding is coming from the same organization, a large percentage of the information on the application is going to be the same across the board.
This is where the common application comes in-- instead of creating a full application for each program with a great deal of overlap from other programs they have created, you can instead create a common application, which you can apply to as many of your programs as you choose. This way, you can have one document with all the basic data that is needed for all applications and then have add-ons for each program that address that program's individual requirements.