In this example we will allow AIM to keep running in the background so you can get text message alerts without having the application open!
Press to launch Cydia from your Springboard.
Press to select Sections from the tabs at the bottom of the screen.
Press to choose the System section.
Press to select Backgrounder from the list of packages.
Press the Install button at the top right hand corner.
Press the Confirm button at the top right hand corner of the screen.
Once the installation has completed successfully click the large Restart Springboard icon.
Now that you have been returned to the Springboard, open an application that you would like to have running in the background. In this example I will press to launch AIM from the Springboard.
If you have logged into AIM before you will notice that it connects to the network
Now hold the Home button down for a few seconds. Notice a message appears telling you Backgrounding Enabled.
You can now return to your Springboard and if someone sends you a message on AIM your iPhone will audibly inform you that a message has come in. AIM will stay running in the background. Press to launch AIM again. Notice you are not shown the connecting message. You will also notice that you have messages waiting!
NOTE**: This can be used for a lot of other applications such as Simplify Media etc. If you have success with any other applications please let others know in the comments.
You can turn off backgrounder by holding the Home button for a few seconds until you see the Backgrounding Disabled notice.