Corrections can be any means of stopping an unwanted behavior but usually people mean a leash correction, which is a quick pop on a leash to get the dog's attention. It's not supposed to hurt, it's supposed to interrupt a behavior.
Corrections have to be determined by the dog's temperament, their age and ability to modify the behavior, and by where they are in training.
Punishment is not like a correction. Punishing an eight week old puppy for soiling on the rug is unfair and counter-productive. Instead we give an eight week old pup a predictable schedule with many opportunities to potty in the correct spot and then we supervise the puppy when he is not in the correct spot and when he signals his need to potty we get him to the correct spot and praise. If we do not notice the signal quick enough, and catch him beginning to potty, we might correct, "Eh! outside to pee pee." to interrupt the process, as we scoop the pup up to take him out to potty. The puppy has to be able to hold his bladder -- if this is excited urination it makes no sense whatsoever to correct. It is not a voluntary behavior. The pup cannot control it.
What you have to guard against is believing you have a hard dog, a stubborn pup, a defiant puppy. Most of the time, when people think this, their dog is medium or even soft in temperament, the opposite of stubborn -- the dog is afraid of doing the wrong thing and shuts down. It looks like stubbornness, but it is actually caused by heavy-handedness with correction, and for a soft dog that can be simply verbal. Pups are not defiant. Defiance is a human characteristic that dogs don't share. If you ever think your puppy is defiant, STOP, take several steps backwards and try to see it another way.
We can correct by leash pops, or by simply changing the tone of our voice, lower and slower. Eh! is easier and quicker than NO! and it is effective. I usually then tell the dog what we do not do: "Eh, don't jump on the Susie!" then I might stop the game or move out of the area so that the pup cannot continue the behavior -- they do learn. "Eh, gentle with my fingers." or "Eh! that's MINE." then I will redirect with the word "Yours." and give them something they can chew on.
Our voice is our biggest asset when it comes to dogs. If we use physical corrections, then we have to get harsher and harsher as the dog matures and for worse and worse offences. With voice, you can give an "Eh!" or bark and "EH! NO!" for serious offenses.
But the focus should not be on corrections. The focus should be on training. Training the dog to do what you want and praising him for doing what you want. Setting the dog up for success and then praising him is so much more pleasant than setting him up for failure and then clobbering him for it -- and a lot of people do this in the name of proofing. I'm not a fan. First you load the praise. You teach little things and give treats and praise for getting them, every time. And then you start phasing out the treats, and you start expecting more for praise too. You motivate the dog to do what you want through praising him.
I have a youngster in with an old spayed bitch. The OSB, is fatter than any dog I have ever had. So, I have had to act drastically. I had to remove the youngster while he eats. Only he doesn't want to eat in the new location. So he runs and yells at the other dogs, and trips around the field, and will lay down with the food dish between his front legs. And not eat. Ugh!. I can tell him, "EAT Oscar!" Nope. Not hungry, you can lead a horse to water. I can get mad. I can be frustrated. Nope, can't eat now, you're mad. I can be exasperated and just put him back in with the OSB and tell him to starve then. Ok. No problem. Wasn't hungry anyway. I can put on a happy voice, "Go on, go eat your foodie." And then when I come back from taking care of the dogs in front, ask him: "Did you eat your foodie???" in a really excited voice. "Show me." And we walk to his dish. Half done gets an "ok." Just a few kibbles left, "Oh good boy!" All gone gets, "What a good boy you are, you ate all your foodie. Yay!!!"
It's changing behavior using motivation and it works. You can get your dog to behave by kicking the dog, beating the dog with the leash, yanking on a prong collar. It works. But, you can also build a relationship with the dog using your voice, praising him for behavior you want, and using a different voice for disappointment, mediocre effort, do overs -- it is still a correction.
The problem with humans is that we zero in on the negative behaviors and want them to stop, and we should really, you don't want them to become a habit. But we forget or ignore when the dog is doing what we want, what we expect the dog to do. The dog doesn't get the positive feedback and doesn't learn that that was right. So he is less likely to repeat it consistently. We think he gets it, because we properly lured him into a sit, once or twice, but he doesn't get it at all, because we haven't reinforced it with plenty of praise, by using the command, and praising the action with the proper timing.
Instead of punishing or correcting an 8 week old puppy for pottying where he should not, we need to praise the puppy for pottying where he should. "Yay! you went pee pee outside." It is so much more fun for you and the dog than: "Dammit Dog! BAAADD!!! why can't you get this?!?"
Go and read up on Marker training. Michael Ellis on youtube and also the Leerburg website has tons of info. IMO it is the best way to train any dog. You never correct a puppy when you train it. Everything is motivational and the key focus is on handler engagement. The No command in marker training acts as a type of correction but it is stress-free and helps the dog build drive and learn to think for itself.
Where you do, however correct a puppy is when it comes to pack/behavioural issues such as trying to grab food off the table, sticking its nose in the fridge when you open it, acting inappropriate around your other dogs/cats/people etc. Even here a sharp No will usually suffice. Puppies need rules and structure but at this young age its all about building a bond with the dog and motivation and encouragement goes a long way.
I always just said "no-no" when I would remove my puppy fro chewing on the kitchen island for instance. Not stern. Just a recognizable cue that he learned. Immediately sticky one of his toys in his mouth.
Also think creatively. I left a bone or an antler at every corner of the island. When he toddling over there to gnaw on it, he woukd go hey! Cool! And chew on his own thing instead.
Finally I figured out that since I was feeding one of my dogs there, tiny bits of food were spattering on the island that I could not really see but puppy's super nose could smell. I scrubbed it and started feeding that dog in the middle of the kitchen.