What's a Internet server?
Well, any server connected in any way to the public internet. It doesn't have to be a Web server (ie IIS) serving html etc, it could be FTP, database, anything that a server is capable of doing communicating on the Internet. The Internet is the network, the server attached can do whatever it likes, nothing makes it a special Internet server other than it is connected. Any server NOT connected to the internet could preform the same services to internal clients (intranet).
What are the uses of a "Internet server"
See above, but the answer is endless (hopefully). I have servers that never see browser traffic. Just DB requests. I have FTP and email servers. These are the common applications of such.
Physical or software.
Both. You must have some sort of hardware to host a server, and that server must run some software to be useful. It will need a NOS of some sort, and something to serve. The question is, what do you want it to do.
How to install.
Buy a server. Buy a OS. Mix in whatever apps you like, poof you have a server.