I am looking mostly for a bit of direction on what Linux OS and/or software is available or best for creating my own "application server" for a home network.

Ideally, I want to be able to supply a Windows/Linux/OS X/other OS application to a remote client, regardless of the client operating system (most importantly Linux and Windows). I don't want to supply a full desktop environment, even though I would also like to do this, I just want to supply the application.

In my mind, it seems like there would be multiple virtual machines with different OS's running on a single server, and applications within those virtual machines are supplied to a client workstation/laptop.

This would operate over a home network and perhaps also over the internet.

I currently know that Citrix does something similar, because I've used it as a client (although I don't know the details behind it), and I am currently using VirtualBox on my workstation to run multiple OS's.

EDIT: VirtualBox actually allows remote virtual machines that are not displayed on the host. This is a step in the right direction. However, any info on hosting applications only would still be helpful, as that is still my main goal.

Conceptually, sounds like you may want this for Linux http://www.ltsp.org/. Note that Linux executables will not run on MS and vice versa, and the exes will actually run on the client.
You'll need to ensure that MS systems only access MS SW and vice versa.
Alternately look into Samba or NFS.