The old favourite stick and carrot method. Well, as a parent of four children who desperately tries to do a good job, I think it is all about finding whatever lever you can that will work. Our friends just told us an excellent example. Their daughter always sucked her thumb with disastrous results. After trying everything, they agreed to buy their daughter whatever she wanted only if she could stop sucking her thumb for 21 days. After the first 5 days of success, they bought her a doll that she wanted but left it on top of the fridge so she could see it but not get at it. After the remaining days she was given the doll, but only on the condition that if she was caught sucking her thumb again she would have the doll taken away. This has happened only twice, and now their daughter no longer sucks her thumb. Finding the right leverage is essential!
I teach in a charter school that uses the positive reinforcement method to train our students (with excellent results). Rather than focus on a particular student's negative behavior, we focus on and reward desirable behavior that we see in a neighboring student, in the hopes that the one misbehaving will see and change their behavior to a more desirable one. This also reinforces the excellent behavior of the student that is doing what they're supposed to be doing!
Remember the words of the song sung by Bing Crosby, "Accentuate the positive! Eliminate the negative! Latch on to the affirmative! Don't mess with Mr. In-Between!"? Well, no truer words were ever spoken!
Negative reinforcement could also involve the return of privileges after they have been taken away, such as no longer "grounding" a child, or allowing television or phone use after they have been taken away. Positive reinforcement can also involve modeling on the part of the parent, perhaps in the proper manners and etiquette at the dinner table, or in social interactions with others.
While the above postings offer wonderful insight into positive reinforcement, I will look at the negative.
Questions about negative reinforcement always remind me of when my father was trying to break me of biting my nails as a child. My father would paint my nails with a biter solution which made me not want to bite my nails. Given that this "treatment" made me wish to not bite my nails ("because a negative condition is stopped or avoided as a consequence of the behavior"--http://www.mcli.dist.maricopa.edu/proj/nru/nr.html).
Time out is a form of negative reinforcement, in that you place the child in an environment wil very little stimulation until the child can calm down and get his/her behavior under control. Then releasing the child from time out becomes the reinforcement for self control. In older children, being grounded can be used in the same fashion.
One way to use positive reinforcement might be to get other persons (in addition to parents themselves) involved in doing the reinforcement. Children will sometimes listen to almost anyone other than their parents, at least during certain periods of their lives (such as the teen years). Getting peers, teachers, coaches, other respected adults, et al. involved in reinforcing positive behavior can, I think, often have at least as much impact as positive reinforcement from parents themselves. In any case, such extra reinforcement certainly can't hurt.
Positive reinforcement can be done in many ways. Verbal praise can be very effective, as was suggested, but there can be other ways as well. You can have a special day with the kids for good grades. You can reward children with special privileges such as a movie, extra dessert, or whatever they like.
Examples of negative reinforcement are harder to come by, but here is an example. How about removing curfew for responsible behavior?
Positive reinforcement is something that parents use all the time with children. This can be something as simple as praising your child when he or she acts in the way you want. So, if you want your child to stop sucking her thumb, when you see her not sucking her thumb (in a situation where she often does suck), you can praise her. This is something she will value and it will positively reinforce her behavior (of not sucking her thumb).
Negative reinforcement is less common since it involves removal of some unpleasant stimulus when your child does what you want. The most common example of this would be removing some behavior of your own that the child does not like. In other words, if your child hates being constantly reminded to clean up her room, the removal of the nagging when she actually does clean her room would be an example of negative reinforcement.