Make a singleplayer world, go into options, and click on "Open to LAN". It should then show up as a server to anyone on your internet. Its a bit buggy sometimes, so google it if nothing appears after a while.
Assuming you are all in the same house (and I think you need to be on the same wifi, but I'm not sure), you can load the world, then press esc to open options, then hit "Start LAN world" or something similar. Then click on it, set the game mode to the desired one in the options, then click start or whatever it is. Then your kids should be able to log in to minecraft in their computer, scroll to the bottom of their multiplayer lists and your world should be there.
You all need to different payed accounts, so you can't all use the same account. Also I think you need to be in minecraft 1.7 or up.
Hope this helps, tell me if you don't understand anything