Yeah-- There seems to be a few schools of thought... and I have STRONG FEELINGS ABOUT IT.
I learned of the concept of allowance from the Brady Bunch, wherein the loving, but stern, Mike Brady somehow managed to pay allowance to six kids while also having a house keeper, a wife that stayed home, and a 4 bedroom house in the valley. It seemed like all the kids had one chore, and got paid a pittance for it.
So I pitched it to my dad-- I'll make sure the garbage is always taken out, you pay me money every week. Dad was thrilled-- said it was a great way to learn responsibility and working for a wage.
That's the first school of thought-- you're teaching kids about working for a living, which we all have to do. You do work, you get paid. Simple as that.
Here's where it went off the rails for me... my dad is actually a really lazy ***hole. As the years went by, my chore list got bigger, and the money never went up. By the time I was ten, he had me doing semi-professional yard work (terracing an embankment in the bank yard) for the same paltry sum I had asked for when I was 6.
Then it got worse. The summers of ages 12-15, technically the last summers I would have had for adventures and freedom, was spent babysitting my little brother, all day every day-- for no money.
By then my dad was raiding my piggy bank on the regular for beer money, the allowance stopped altogether, and I was told that I shouldn't have to be paid for spending time and helping out my own family. Asking for money became a way for him to guilt trip and manipulate me into doing stuff he didn't want to, or could afford to have done, himself.
So all that taught me was that it was okay to rot in a job for years without getting a raise because the company needed you. Which I did in every job I had between the ages of 20 and 30.
That leads to the other school of thought, which Reese has said, that you don't have to tie chores to a reward system. You want them to understand that contributing to the responsibilities of life is a thing we must all do. As Met said he experienced, you do the work, and your parents may gift you an allowance for going along with it and not screwing up.
I think this approach if you have a kid who genuinely wants to be helpful. Not every kid is like that. I had a cousin who was a born troubler-maker. Nice guy. really funny, but just loved being in the middle of **** constantly. As a kid he was always in trouble, and he had zero interest in chores or helping out in any way. I'm sure there are lots of psychological reasons he turned out that way, but the point is-- you can't expect a kid to want to help without some incentive or training.
We've ended up taking a bit of a hybrid approach with my kid.
When he was 5-8, we would give him a single chore, that paid a small amount. It was just to establish the notion that sometimes you have to work for what you want.
From ages 9-12 we started being stricter about cleaning up after himself around the house. Keep his room clean, take out the garbage in his room and bathroom, wash his own dishes. At first he wanted to get paid for this-- because he was getting paid for the kitchen garbage and he had friends getting paid for chores. We told him "no way." These things aren't a job, these are chores you have to do as part of a member of this house. The kitchen garbage, THAT'S a job. Your room, is your responsibility.
Through middle school taking out the kitchen garbage fell off, which is fine. He got it. He just started doing it when it needed taking out-- which by the way is the opposite of me as a kid. I had to take out the garbage got bad, he'd do something about it. We stopped the allowance at this stage because anything he wanted money for was not going to be covered with the chump change we'd pay him. Any time we'd need help around the house with something extra, I'd kick him a few bucks for helping out. After a couple years though, I stopped doing that. I also stopped asking for help-- and it turns out, he realized he is happy to help. He'll always stop what he is doing to help with something if I need it. Because I didn't make him waste his time as a kid doing chores I was supposed to do (like my dad did), now he just wants to help for the sake of helping. Toward the end of this stretch he wouldn't never expect money for helping out with anything in the home, but on occasion, if he needed money he'd ask for a special task he could make money for. THIS is the stage at which I'd assign him some sort of heavy labor that my dad would make me do. Work for money on a task.
Now that he's in high school, the only thing he does that I have to get after him about is not leaving dirty dishes in the sink. I don't give him any. When he needs money, he'll still ask for a big task... but there isn't always one to do. He also understands now that I pay for his phone and I don't lock it down, which he appreciates. All his friends have their phones locked down and he feels pretty lucky. I know those parents feel they are protecting their kids, but I feel like restrictions are a form of punishment. If he screws up, THEN I'll lock it down. But I monitor what he does, no need to rob him of agency.
Anyway... money isn't a huge issue. He doesn't have a lot of needs. He always collects a bunch of cash on his birthday and Xmas, and he dumps into his amazon account and unlike me, he doesn't feel the desire to spend it all at once. When he wants something, he'll order it, and it shows up.