Find some bin bags and old towels. Put the bin bags down on the floor, towels around the pipes. Hope you haven't got a carpet you like.
Easier with two people - you can close the valves on the radiator (make a note of their current settings so you can put it back the way it was - count the turns on the lockshield, look at the number on the TRV). Let a bit of pressure out using the bleed valve so that water doesn't spurt out when you undo the fittings (don't forget to close the bleed valve up or you'll find out just how quickly a radiator can drain). Then, undo the fittings, stick your fingers in the radiator holes and then flip the radiator upside down which should stop it pouring all its contents out over your floor. If you can, get the radiator outside whilst you figure out your next step - doesn't matter how much it leaks out there.
When I did this, I found the valves were still leaking a bit so I ended up draining the system anyway. To stop the leak I just put some insulation tape over the open pipe which obviously only works if you've relieved the pressure on the system (it was already draining via a hose at this point, so no pressure to deal with). If you get a leak you'll want to make sure that no additional water is getting into the system - if it's a vented system you'll want to be turning off the valve coming from the feed & expansion tank, or if it's seized like ours was you'll be wanting to either turn off the mains or tie up the ball valve.
If your system hasn't been drained or flushed recently expect the water to be black and stainy.
Whilst you've got the radiator off the wall you may as well pour some water through it using a hose, give it a nice shake and shimmy to get as much gunk out of it as you can before putting it back on the system.
If you end up draining the system, you'll be wanting to add some corrosion inhibitor when refilling - this is easy if it's a vented system with a feed & expansion tank in the loft but otherwise you'll need to add it via a radiator valve or something. In our case we have a Magnaclean filter so I drained that and added the inhibitor there.
So, easy in theory but there's a ton of bullshit things that can go wrong.
Oh, and older radiators can be surprisingly heavy.