Yes the technology is called a Virtual Tabletop or VTT. It combines the following
- A whiteboard that everybody can draw on or use to display image. Also used as a battleboard with token instead of miniatures.
- a voice and/or text chat engine
- RPG software utility usually including a dice roller, character sheet, and random tables.
Popular VTTs include
This perhaps the easiest to try and get started with it. It can be used for free and has a subscription option that unlocks more features and increased on-line stores of maps and images. The features are centered around customizing RPG software utilities, and sophisticated whiteboard features like dynamic lighting for maps.
This is a traditional piece of windows software that you install. The referee will act as a server so this will require a little bit of messing around with your home router. The players will act as clients connect to the referee. There is no voice, participants usually use Google Hangouts, Skype, or other VOIP utilities. Fantasy Grounds has a subscription service and a one time purchase option with lifetime updates (to day). There are three levels of both, player only. Referee which can only have those with the player client connect. This is also doubles as a player client. Finally Ultimate which allow the demo version of Fantasy Grounds to connect to the referee as a player.
FG v. Roll20
Fantasy Grounds is has the most sophisticated software utilities of any VTTs. Partially because the programmers of FG have free reign as it is a traditional application. Roll20 is constrained by the web interface. However because of overall advancement in web technology is it rapidly becoming a non-issue.
Roll20 has a better social site surrounding it's VTT. Given your situation you will be able to browse the list of open games and pick one. Or you can post your own campaign and recruit people to join. Both have marketplaces where you can buy add-ons like tokens, maps, and most importantly rulesets for specific RPGs. Some are free and some are licensed. Fantasy Grounds has the Official D&D 5e license if that makes a difference for you. Fantasy Grounds is available on Steam which makes it convenient to buy and install.
Fantasy Grounds has the best dice rolling utility of all the VTTs. It looks and feels like rolling real dice.
There are other VTTs but that like saying there are other OSs for a phone/table than iOS or Android. Yeah they exist but those two are the two most important out there.
For a good list go to RPG Virtual Tabletop. They have a lot more information on the other VTTs than I can type here.
For a breakdown of games and the number of people playing on Fantasy Grounds look here.
For Roll20 look here. They release one every quarter.
So does it work?
Yes it works very well. It is playing tabletop RPGs over the internet. You use the same stuff, do nearly the same prep as you do with face-to-face. Some groups, including the one I game with, freely switch between FtF and the VTT. We play weekly on Roll20 and get together monthy for face to face gaming.
The downside is twofold, one it is a form of chatting over the internet like using Hangouts, Facebook Chat, or Skype. If you don't like this type of interaction then none of the VTTs will cut it. Second anything that you want to show you will need to get into the computer either as text or image. Now with Google search and inexpensive scanner it not hard to find stuff to use or get your own stuff. But it is the major difference between playing a VTT based campaign and a face to face campaign.
There is one significant upside to VTT is that a VTT can greatly enhance the use of miniatures (as tokens), maps, and battleboards. Nearly all VTTs have a straight forward method of handling fog of war and some do line of sight (like Roll20) as well.
Other ways of playing RPGs over the internet.
First off VTTs are the only way that is found to date that replicates how we play tabletop RPGs face to face on the internet. There are other ways of playing roleplaying games on the internet but they all involve compromises from how a game is run face to face.
There are MMORPGs and CRPGS like World of Warcraft and Skyrim where the computer is substituted for a human referee. Some are multiplayer, some are standalone.
There are play by post that unfold at much slower pace than a face to face tabletop session. However they are much more convenient to participate in for some.
You can use just voice/chat however the inability to see the dice and/or images makes the campaign less personal in some ways.
I can't stress enough that when it comes to VTTs it is playing tabletop.