How to make a simple Video Switch?
You are right in thinking you can use relays.  However, you are wrong in thinking you can switch just the ground and leave the signal connected.

The shield round the RCA cable is only one of the ground routes between the TV and the video source. They are also connected via the mains cables, unless you're running from batteries.  With the ground disconnected as you were envisaging, the signal would return to the video source through another route, one which is very noisy, high impedance, and just generally messy.  So your signal would still be there, but massively corrupted.

Secondly, as that signal will still be there, so will the other signal, so you will have both signals there all the time fighting with each other.

A relay, as commented, is a mechanical switch.  It is an electromagnet that moves a chunk of metal (armature) and forces it against one of two metal plates (contacts).  The "common" connection is linked to the armature, and the "NC" and "NO", or "Normally Closed" and "Normally Open" pins are connected to the two contacts.  A spring returns the armature to the "NC" position when power is removed from the coil.

So, with 3 relays, you can have 2 groups of 3 signals switched.  The TV end goes to each of the common pins, and the incoming video signals go to the NC and NO pins, one for each video (make sure you keep them the same for all the video signals ;) )

As you only have 2 relays that is a little tricky.  I can tell you now that it isn't possible to switch 3 signals with 2 switches.  So you need to add a third relay.  You could either buy 2 boards and control them individually, or you could get 2 more extra relays that run from the same power and use one of the on-board relays to switch the other two relays.  Or to simplify, those two extra relays could be one single "double pole" relay - a relay that is like having 2 relays in one box with one coil - it switches 2 signals at once.