Click on System -> Administration -> Synaptic Package Manager
This will open Synaptic which is networked repositories of all the programs usually used by Ubuntu users. If you have the gnome desktop you should try to stick to gnome (gtk, gtk+ or gtk2.0) programs. If you use KDE then it gets a little confusing from my point of view. It is kpackagekit, kynaptic or adept depending on which version of KDE you have! It couldn't be more stupid. Add to that that most of KDE's programs say they are for KDE3 or 4 or begin with the letter 'K'. (look at wikipedia for a simplified view of why and when changes happened)
If you see files that end in .deb they should install like a windows executable using a program called gdebi. They are quite common on sites where different systems are supported. And at getdeb.net there are a few you might like.
If you can use the command line which means opening 'terminal' (though you type 'gnome-terminal' if you Alt+F2 and in gnome-do)
As I said some sites will point you toward a ubuntu build of their software. But I should add that you should only go to a software's own site as opposed to someone else who is hosting a build because they could slip a virus in. This is the beauty of Synaptic. It is trusted software.
Compiling a package isn't that difficult but shouldn't need to be done often. (you may think get the very newest build is the best but it might not work with your system which why if it is in synaptic use that version). There will often be instructions in a file called readme or install. There are several different archiving and compression methods ie .tar.bz .zip .tar.bz2 .tar.gz2 etc and these need a different command or argument/switch when using tar which you can read about in the man page (type 'man tar' or 'man gzip')
So most often you will do this:-
tar -xvvzf foo.tar.gz for a file named foo
./configure
make
make install
(before doing this install the tools like this:
sudo apt-get install build-essential
usually I've already opened synaptic and installed g++ gcc gfortran - tools you need to compile your own programs in c c++ and fortran though the latter I only tried to see it work)
And before doing even that - Look for the program authors build instructions as the above 3 lines often change a little!
Even if you don't use the last instructions you now have an idea of what is involved. They are changing the packaging system soon I read. Hope this doesn't leave us as confused as KDE users must be.
Good Luck