There are several different French sauces that begin with the simple brown sauce recipe given here. Brown sauce or sauce brune is also known as Spanish sauce (sauce espagnole). It is made from roux (flour and butter mixture) which you brown for a few minutes before adding the bouillon.
Create Your Own Brown Sauce
You can try cooking various finely chopped vegetables (carrots, mushrooms, celery, shallots, etc.) in the melted butter before adding the flour to get different effects. Sometimes bacon is also used in a brown sauce.
When vegetables are added at the start, the French often strain the sauce at the end to get a homogenous fluid. You might find this step unnecessary. I do! This brown sauce and its variations are typically served with meat, but some go well with poultry and braised vegetables.