Can I run multiple virtual machines at once?

Yes you can run multiple virtual machines at once. They can appear as separate windowed applications or take over the full screen. You use one keyboard/mouse. Just as with other applications when the virtual machine has 'focus' it receives the input from the keyboard/mouse.

The hard-and-fast limit to the number of VMs you can run is your computer's memory. However, the more virtual machines you have, the slower everything will get - in particular, disk I/O is typically the first and worst performance bottleneck you'll face, since most disk performance degrades much worse than linearly with increased concurrency.

It looks like this software was made by windows for windows, correct?

VirtualPC was originally developed by Connectix and later acquired by Microsoft. In addition to the version that lets you run Windows VMs on a Windows host, Connectix also had a version that let you run Windows VMs on a Mac host. Some competing virtualization products are VMware (Windows, Mac, Linux), VirtualBox (Windows, Mac, Linux), Parallels (Mac), Xen (Windows, Linux), and KVM (Linux).

With a virtual machine, can I change the resolution size of a windows virtual machine, so it thinks it is displaying information on monitor of a given size?

Yes. In addition to running in windowed mode, you can also run in full-screen or in a "seamless" mode where the VM's desktop is not visible but applications running in the VM show up directly on the host machine's desktop. Different virtualization platforms have different names for the "seamless" mode.

Can I run multiple virtual machines at once?

Generally, yes

Do they each have their own mouse? How does that work? I move my mouse over the virtual machine window and suddenly I have control of that mouse? Do my keyboard actions only correspond to the main PC, or once I click on a virtual machine window, do all mouse and keyboard input get redirected there?

You have 1 mouse cursor which can seamlessly move in and out of the VM window (if in windowed mode), assuming you have an integration utility installed in the VM. If you don't have the utility installed in the VM, you can click in the window to let that VM grab control of the mouse cursor and keyboard, then press a special keystroke (usually the left Alt button, or Ctrl+Alt) to release the mouse cursor and keyboard back to the host OS.