Ubuntu is based on Debian - .deb format for software packages / Fedora and others are .rpm packages. Ubuntu's Software Center has preloaded repositories for software packages that will work for the most part on the OS. One can also install packages from websites like skype and Google Chrome by downloading .deb, which the software center can aid in the installation.
apt-get (advanced packaging tool)
search for a repository: apt-cache search python
search and scroll: apt-cache search python | less
add new repository (example from ubuntu touch on nexus 4): sudo add-apt-repository ppa:phablet-team/tools
update package list: sudo apt-get update
install: sudo apt-get install packagename
remove: sudo apt-get remove package1 2 3....
add sudo apt-get remove -purge option to remove config files as well
sudo apt-get autoremove will remove any unused packages
upgrade: sudo apt-get upgrade packagename
sudo apt-get upgrade upgrades everything
sudo apt-get dist-upgrade will upgrade anything that has a slightly different name than the old version.
clean up: sudo apt-get clean to clean up caches from downloads
sudo apt-get autoclean to clean up cache but keep the newest versions for your upgraded packages
check: sudo dpkg -list | less to see a list of what's installed.
easter egg: sudo apt-get moo
3rd Party Repositories
Often they may be automatically add to the Ubuntu software repositories list from a parties website with their .deb file which lets it update through Ubuntu.
Personal Package Archives
Anyone can create a PPA thru Canonical's Launchpad System. Ex. PPA for Nexus 4 Ubuntu Touch software package.
Use at your own risk.
Once added, can install thru apt-get or Software Center, etc.
Compile from Source Code
Packages compiled from source. Generally you won't need to do this, most packages are done for you.
Usually show up in .tar.gz's.
Install Build Essential package (sudo apt-get install build-essential) for software used for compiling.
Find source code from the github or wherever you're downloading it from. Probably in the .tar.gz file.
Extract w/ tar -xzvf file.tar.gz or tar -xjvf file.tar.bz2
Change directory to the new directory created for code.
Run ./configure (or check readme or install file if that doesn't work)
If you get errors, run apt-get install for any packages needed.
Repeat configure command til completed.
Can also use sudo apt-get build-dep for new versions of programs already in Ubuntu's repos.
make command compiles the program.
sudo make install to install the program.
sudo make uninstall to uninstall the program.
Closed source software may be in binary form, in which case you would download the .tar.bz2 etc, extract the archive, and run the .sh script.