Béarnaise frightens people because it is delicate and it can break. That reputation is fair - it is a warm emulsion made over gentle heat, and carelessness with temperature ruins it. But the technique is learnable in an afternoon, and once it is in your repertoire, a pan of steak on a Tuesday night becomes something worth sitting down for properly.
Béarnaise and hollandaise use the same emulsification technique - egg yolks and fat, held together over heat. The difference is in the flavour base. Hollandaise uses lemon juice as the acid; béarnaise builds its acid from a reduction of white wine, white wine vinegar, shallots and tarragon. That reduction is cooked down to concentrate its flavour before the eggs go anywhere near the pan. The result is more complex, more savoury, and more distinctly French than hollandaise.
The reduction is where the flavour of béarnaise actually lives. You are making a small, very intense liquid - a few tablespoons - that carries the shallot, wine and herb character into the finished sauce. Do not rush it or skip it. If you are working without fresh tarragon, dried is a passable substitute for the reduction; use fresh at the end if you can get it, as the raw tarragon added at the finish is what gives béarnaise its characteristic bright anise note.
For the reduction:
For the sauce:
The classic pairing is a grilled or pan-seared steak - rib-eye, sirloin or fillet all work. Beyond steak, béarnaise is excellent with roasted lamb, steamed asparagus, poached salmon and fried potatoes. It is an assertive sauce, so the food beneath it should be able to hold its own.
If the sauce starts to look grainy or the egg begins to scramble, you have overheated it. Remove it from the heat immediately and whisk in an ice cube or a tablespoon of cold water - this sometimes saves it. If the sauce breaks into an oily mess, start with a fresh yolk in a clean bowl and very slowly whisk the broken sauce into it, as you would with split mayonnaise.
The most important habit: keep the heat low. The eggs should thicken slowly. If the water below is boiling vigorously, reduce it to a gentle simmer.
Béarnaise does not hold or reheat well. Make it close to serving time, keep it warm over the hot water bath, and plan to use it within an hour. For a full overview of warm and cold emulsion sauces and how they relate to each other, the complete guide to homemade sauces and condiments is a useful reference.