I currently have one very big partition in my laptop that runs Ubuntu. I have to install Fedora for work and I'd also like to try out OpenSUSE, so I'll have to repartition. Since I don't want to duplicate data, I will move /home to a different partition and mount it from all three. I'd like to know, can I also do this with /var and /usr? If so, would that mean that every program I install will be available from all three?
No, you cannot. Programs/libraries in /usr could be different for each distro and may not be compatible as is. Also do not share. Likewise, /var cannot be shared as different versions of the same program that are present in different distros may treat the data in /var different.
I HIGHLY advise you do not do this if you want to maintain a working system.
If you need to understand what each directory does in general, please see this Wikipedia article
Yes you can, but don't. The main point of having distributions, and not working with Linux raw, is that the people who put the distributions together have figured out how to have all these programs working together alongside each other. As soon as you try to mix and match between them, you are going to be having to figure all this stuff out for yourself. And the packagement management systems of the distributions will suddenly become your enemies.
Having said that, on my Mac Os laptop, I have been using two package managers alongside each other: the ghastly Fink and the equally ghastly, now defunct DarwinPorts, and it sort of works, because both of them take a little bit of care not to touch on each other's feet, or interfere with the Apple installation. And they do that... by mainatining their own /usr and /var hierarchies, under /opt and /sw. It's crap: I have three installations of Perl 5.8, and it's all I can do to make Fink believe that I have a 5Gb Tex Live installation and not add another Tex installation of its badly configured own.