Melt the butter and keep it warm.  

Heat the vinegar or lemon juice until just warmed.  Have small saucepan with boiling water and a measuring tablespoon ready. 

Place the top of a double boiler over (not in) hot water. (This means the bottom of the top of the double boiler sound not make contact with the water heating in the bottom half of the double boiler.)

Place the egg yolks in the top of a double boiler and whisk until they begin to thicken.   Now add 1 tablespoon of the boiling water.  Continue to beat the sauce until it begins to thicken. Repeat with the remaining water, one tablespoon at a time, beating the mixture after each addition.

Now add the warmed vinegar or lemon juice. Remove the double boiler from the heat.   Beat the sauce briskly with a wire whisk. Continue to beat the mixture as you slowly pour in the melted butter.  Add the salt and cayenne and beat the sauce until it is thick. Serve immediately.

Your Sauce Separated, Now What?

1.  If the sauce starts to separate, add 1 or 2 tablespoons of cream and beat the sauce with a wire whisk until it is smooth once again.

2. If the sauce has curdled, you can put it in a blender and blend it.   This will alter the texture a bit.

History of Hollandaise
Alan Davidison states one of the earliest recorded versions of the sauce dates back to 1758 "sauce a la hollandoise" from Marin's Dons de Comus.  This recipe included butter, flour, bouillon, and herbs; no egg yolks. Davidson also quotes from MeGee (1990) who explains eggs are not needed at all and proper emulsification which can simply be done with butter.  He also states that if one does wish to use eggs they are not needed in quantities normally called for in traditional recipes.

Other Sauces derived from hollandaise include:
sauce aux capres - add drained capers
maltaise - add blood organges
mousseline or chantilly - addition of whipped cream
moutarde - with Dijon mustard