What can VPS do for me and how I can operate that remote system?

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Dear myLot-ers. Somehow I obtain information that there is such service as VPS. I start to gather information about it but after some hours of reading I still cant get the idea what exactly can VPS do for me and the most important thing, if I try one, how can I operate it? My friend simply said: "It is like your own computer connected to the internet turned on 24h/day and you can do anything with it like you will with your own including installing programs and running it on that machine. You can control it remotely." It is clear to me that I can transfer files to and from that virtual PC but theoreticaly can I transfer instalation files of program to that VPS, then install it run and start using that program? (if it will be usefull) And how can I control whole proces of installation and running it? Because I dont get it. While I am using programs on desktop PC I see GUI on monitor but how can I control when It runs on other system? Any kind of visualization is done or what?... And the last thing, like I know there is free web hosting services, is there any free VPS services where can I try it on my own? Sorry for maybe dumb questions, but I will be thankfull if you can enlighten me.