2 network ip addresses, one computer...help!

You need a second Network Card for the other network.

Grumpy

If you put in your own router, most likely, your IT folks will neuter you. (I might be biased, I happen to be a hospital IT guy).

I agree with Grumpy, get a second NIC

It is one of the IT guys who is unofficially telling me to go ahead and just hook the system up to the hospital's network.

Anyway, the pharmacy system has a box that has about 8 ethernet ports. (I do not know if this is a hub or a router). What I did was take an ethernet cable plugged into the hospital's network and plugged it into this hub. Then I ran 2 ethernet cables to 2 different machines...one is the hospital network PC (for my work related emails, wholesaler ordering, etc.) and another ethernet cable going to my pharmacy system for when I need to get on the internet for tech support from my pharmacy system vendor.

I don't know if explaining it in this detail makes a difference as to what I can do, but another way of asking my initial question is this:

Can I configure my PC to have 2 different network connections (at the same time) and make one the primary and the other secondary, and just change which one I want to use?

If you dual home your system you run the risk of exposing one network to the compromise of another. Any systems administrator worth their weight would never allow you to dual home your system. Any system that has two nics, once compromise, will compromise both networks.

The greatest threat is internal.

There is no risk of compromise as my pharmacy system network is internal with 3 PC's and a cash register. There is no internet access, but we do have a secure T1 line for our insurance claims and credit card transactions. Our T1 provider is just a data relayer and they do NOT provide internet access.