People come to barre expecting something dainty and leave walking down stairs sideways. Watching it happen to strong people is one of my reliable pleasures.
Why small movements are hard
Most training moves a joint through its whole range. Barre parks you near the end of one and asks for an inch, then another inch, then thirty more.
Working in a narrow range with no rest keeps a muscle contracted continuously, and continuous contraction is what produces that specific burn. It is a different demand from lifting something heavy, and being good at one predicts very little about the other. Cyclists find this out fast.
Where it comes from
Ballet, by way of a century of people borrowing from it. The barre itself is a ballet training tool and the position work is recognisably balletic, but a barre fitness class is not a ballet class and does not pretend to be. If ballet is what you want, I run that separately.
What has carried over is the attention to alignment, the turnout, and the willingness to do something very small very many times.
The bits that catch people
Outer hip. Nobody expects the outer hip, and I never warn people, because the look on their faces the next morning is half the fun. It works quietly all class and it will let you know that evening.
The other one is your standing leg. Half of barre is balance, and the leg you are not thinking about is working harder than the one you are.
If you are coming from Pilates
You will recognise the precision and the vocabulary. The pace is different and so is the intent: Pilates is generally after control through a range, barre is after endurance at the edge of one.
They complement each other well. Doing both in the same week is not duplication.
Practical
Grip socks if you own them. Expect to shake, and understand that shaking is not a sign you are doing it wrong. Everybody shakes. That is what the class is.