Oh, your 'package manager' is going to be like the Add/Remove Programs thing in Windows, and the 'repository' or 'repo' is basically a directory on somebody's server where they keep these for people to download.
A 'package' is some compressed form of the program you want to install. Linux programs are more modular, so a program will almost always depend on some others. The package manager has to resolve these dependencies and the correct versions of everything it needs. You could do it by hand, but that would be masochism. For example, my command-line tool is called 'zypper' and it installs packages called 'RPM's (after Redhat Package Manager). So when I go to install something, it'll do something like this:
hermes:~ # zypper install mpg123
Loading repository data...
Warning: Repository 'OpenSUSE-12.3-Update-Non-Oss' appears to outdated. Consider using a different mirror or server.
Reading installed packages...
Resolving package dependencies...
The following NEW packages are going to be installed:
libmpg123-0 libopenal1 mpg123 openal-soft
The following recommended package was automatically selected:
openal-soft
4 new packages to install.
Overall download size: 378.3 KiB. After the operation, additional 803.4 KiB will be used.
Continue? [y/n/?] (y):
It's going to download all the software libraries and other crap it needs.
The package manager for Debian and other distros that're based on it, like Ubuntu and Mint, is called 'apt', and it's a set of programs so the installer is called 'apt-get'. But the GUI will have a GUI version so you shouldn't have to use the command line. Some of the GUI front-ends are really nice, some aren't. If I wanted to install mpg123 (a command line mp3 player, so almost useless but just an example) on Mint, I could open a shell and type
computername# apt-get install mpg123
It'll be downloaded and installed then.
Because of all the different versions and dependencies, for some software like music and video players e.g., it used to be the case that it was sometimes hard to find the dependencies. For example, MPEG is a commercial codec so maybe you had to find a third-party repository that had the codec software that your music player needed, and the version wouldn't work with the library versions you already had on your system, so you ended up in 'dependency hell'. If you stuck with OGG though, a free codec, you wouldn't have the problem. So you'd only be able to listen to music you'd ripped from your own CDs. ;)
Ubuntu doesn't have problems like this so you'll be fine.
Oh, the 'shell' is like an actually-useful Command Prompt. You can use it to basically do anything you could in the GUI. You can start LibreOffice Writer by just typing 'oowriter', edit text files, it even has it's own built-in programming language. There are choices in that regard, too, but most people just use the 'bash' shell. Only a real nerd would use ksh ;) The icon or menu item will be called 'Terminal', 'Shell', or 'Bash', but sometimes it's a more advanced program like KDE's Konsole. Usually, you can right-click on the desktop and open a shell.
The commands usually take some options, which can be just a '-' followed by a letter (Unix style) or, for better remembering, '--' (two dashes, GNU style) and a word. So, to list some files...
[email protected]:~/doc/test> ls
file1 file2 file3
[email protected]:~/doc/test> ls -a
. .. file1 file2 file3
[email protected]:~/doc/test> ls --all
. .. file1 file2 file3
ls is like dir in Windows, and ls -a lists all files, even hidden ones, which are called 'dot files' because to hide a file all you have to do is make the '.' the first character of its name. The single dot with no name represents the current directory, the double dot the parent directory.
The other thing is that, Linux is a multi-user operating system, so it's very secure in that regard. The user that's called 'Administrator' is called 'root' on Linux. You can use the 'su' (switch user) command to become root to install programs or do other privileged tasks, like system-wide configuration. In Ubuntu though, and maybe Mint, they'll want you to use 'sudo' which is like a one-time version of su so you're only root for that one command, though it does remember your password for a while.
The absolute, most important command is 'man', and then 'apropos'. Most software, including the operating system, installs what are called 'man pages' which is short for 'manual pages'. I couldn't do my job without man ;) If I want to search the manual for pages about something, I use 'apropos', then find the keyword and "section" I need. So, for example:
hermes:~ # apropos zypper
zypper (8) - Command-line interface to ZYpp system management library (libzypp)
zypper-log (8) - Zypper logfile reader
hermes:~ # man 8 zypper
That would display the zypper page from section 8 of the manual.
Eventually you learn the joke...
hermes:~ # man woman
No manual entry for woman
hermes:~ #
Not sure if that's helpful, but maybe as a little overview. Hopefully you're not more confused than before... :)