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Alright... well you know, I've only started playing 3 months ago, but I got a guitar with a Floyd and because I had read about them, I had NO problems whatsoever. It's all about knowing about them. So I made this little guide thingy. It's not complete (it's not like I have loads of experience with it...) and if some of you are willing to make a 'super Floyd guide', I'd be more than happy for this to be the basics for a more complex and complete guide. So suggest and criticize away
The Floyd Rose is a floating bridge. It is not fixed to the body, but rests on two poles. The balance is achieved by countering the string tension with springs. This brings to the rule number 1: unless you have A LOT of experience, do NOT change the string gauge by yourself: the tension will change and your neck will suffer.

The Floyd system has two units: the bridge itself and the locking nut. The bridge locks the strings too, that is why a Floyd and its many licenced derivatives are considered a double locking tremolo. It ensures tuning stability after abusive use of the whammy bar. Judge by yourself: My licenced cheapo Floyd is 17 years old and stays in tune PERFECTLY even after a good dozen of full divebombs.

But of course there is a pay off: The bridge has to be stabilized all the time you tune and it's a hassle. This means that Floyds don't like alternative tunings. Each change in tension in one string changes the tuning of others.

Make sure you clip out the ball ends. They won't fit the Floyd. Or else put them through the machineheads but it looks ugly

Stringing
Assuming you want to change strings, there are 3 ways to do it (Make sure you clip out the ball ends. They won't fit the Floyd. Or else put them through the machineheads but it looks ugly ):
1) take all the strings off, restring, tune. It takes 2 hours, it's hard, and a pain in the ass. Don't do it unless your Floyd is not recessed and it won't back into the body this much. But even so, it's fairly annoying.
2) replace the strings one by one. This method also takes a long time, though less than for 1) and is fairly easier to tune. Though of course, the new strings won't stay in tune for a bit. What I did when I got new strings was doing ****loads of divebombs to stretch the strings out. Then I tuned it again as I'd do normally.
3) block your tremolo. Use anything between the bridge and the body so that it doesn't sink into the guitar. Then, change your strings all at once or one by one.

Tuning
Ok... so your strings are on the guitar, they are stretched though not tuned? Now you need to tune...
1) ensure the strings are properly locked in the bridge. Use an allen key to tighten them if necessary. (I'll add pictures later)
2) that's when problems start... People have different ways of doing it. I mess about a lot with the spring tension at the back so tuning is also done differently taking care of this. Let's just assume the spring position, string gauge and wanted tuning are constant. The way I do it:
Usually the locking nut is unlocked because the fine tuners don't have such a big range. I tune my low E to proper pitch using the mahineheads, then I do the high e, then A, then B, then D, then G. And I repeat it. All the way. Usually it takes 3 to 4 times doing this. When the guitar is in tune, I simply lock the nut and use the fine tuners to finish.

Changing Tuning
You want to drop D?
Some people would unlock the nut for the E and A strings, drop the E, lock again, fine tune and here you go. Well unfortunately... Your strings will still be relatively in tune with each other, but they won't be perfectly in tune.
As Damage mentioned, open tunings really won't work well. I wouldn't try them. I haven't yet myself. I usually keep mine in E standard but right now it's in Dropped C. Here's what I had to do (it still is tuning. Tune with the locking nut unlocked, and when you lock it again, fine tune)
1) I tuned all the strings a full step down.
2) I slackened the spring claw at the back to re-establish balance
3) I tuned again
4) I slackened again. By now the bridge should be in the position you want it. It may take more, depending on how you do it and how you want it.
5) I dropped the D, and tuned it all over again a few times until it was perfectly in tune.

A word on springs
The more springs the more tensions. People most often use 3 springs in a configuration. Next comes the /|\ configuration. It makes bending the lowest and highest strings a little easier. To be honest, I didn't notice much difference.
Some people use 2 or 4 springs. I don't see why not. But to me, 3 springs is perfect. When you put on heavier string gauge though, you may want to put one more springs, as it otherwise you will need to tighten the claw all the way against the wall and it will tighten the springs too much.

Oooooook I think that's it. To everyone who say that Floyds are a hassle: I have just started playing guitar. This guitar pictued here is my first electric guitar. I never had a problem with the Floyd. You just have to know what you're doing before getting one, or learn about it before tweaking it.
Though you do have to be patient

Nice Job... I´ve been working on mine (guide) on and off for about a year now, it just totally blew apart the spectrum I had planned. I think when I´m done the only thing I won´t have taught people is how to mount one... (although come to think of it).....

BTW, ehile the JT-6 may be an Import bridge made of strange materials, it´s probably the only quot;Licensedquot; copy (other than a Schaller or a Gotoh) which I consider adequate to good, it´s just a bit different (in multiple ways) from an OFR

My JT6 is amazing! It really stays in tune well. I see no reason to change it (though I may get a very cheap deal on a Kahler Floyd type (Steeler or Sypder) and for that price I may as well try it) I've had no problems with it. It's just the basic rule of Floyds, they all are annoying. No matter how patient you are, if you compare them to hardtails, they're a pain dans le cul.

Nice work. I'll just offer this: if you are going to be changing tunings much I'd either block-off the Floyd (make it non-recessed) or use a different guitar without a Floyd. I blocked-off two of my guitars and it is a JOY tuning them compared to a regular floating trem.

I have been using Floyds since '88. One trick (or technique) I do 100% of the time (on floating models) is this- when making tuning adjustments I first play each string and watch the tuner to see where I am at. Then I tune from E to E, then A to B, and then G to D. I am usually tuning the E stings one-quarter to one-half step sharp (or flat depending on the weather) knowing the other 4 strings will change the tension of the brige and pull them into tune. That is the ticket! I very seldom have tuning issues.

I think I do more or less what you do too, but without realising it, just out of experience. I didn't really know how to explain it. I am French (though 100% fluent at English) and sometimes I tend to use 100 words when 20 would do
I have only one guitar myself (and I knew what I signed up for) and I do want to block the trem but I am new to this country and I have no idea where to find the necessary material. This summer or next year when I settle into my new flat I'll work on it but right now blocking it would be an unecessary hassle (well as if tuning wasn't already...)
I also made this guide strictly for tuning and a bit of stringing. That's why I asked for things to add

looks good.

it is a good idea to inspect the knife edges and posts from time to time as well. esp. if you're having tuning probs.

and a soft bristle toothbrush is the best thing I've found for cleaning all those knooks and crannies... and that little strip of the body between the bridge and the B pickup.

nice guide.

i got my 1st floyded axe a couple of weeks ago and i dont understand the people who say they're awfull. give, i havent had the need to restring it, but i cannot see it as such a problem