I'll step you through each method.

1. apt-get uses the Debian PPA system to install software. It is what goes on in the background when you install software from the software center.


2. dpkg is even further in the background. It is what Ubuntu uses to actually install software, as opposed to simply managing it. You should not use this method if you can help it, since higher-level methods, such as apt-get, are superior in utility.


3. Installing from the Software Center is the highest-level method of package installation. It is not designed for advanced users, who tend to prefer apt-get. However, it makes the Ubuntu ecosystem more accessible for the general user.


4. Synaptic package manager is like a GUI alternative to apt-get. For apt-get, you need to know the name of a package. Synaptic instead lists packages that your computer knows of, and allows you to install those by clicking on them. It is more advanced than the Software Center and still has the same usage of apt-get.


5. Adding links to sources.list is a way to add a PPA that may require a key. This is the method for installing some proprietary apps like Google Chrome. Doing this does not actually install the app, you still need to update your sources, then perform apt-get install or use Synaptic afterwards. Like Sparhawk said, these can also be used for official repositories, whereas PPAs cannot.


6. PPAs are needed to update software. They are generally what is used to install as well. They are also the main reason apt-get is superior to dpkg for the installation process. apt-get can provide you with updates and dpkg cannot. As Sparhawk said, it is important to note that PPA stands for Personal Package Archive. These are not official repositories, which instead belong in sources.list.