Hi - I read this (I did not write it) and I thought it was worthy of reposting. Excellent content.
Random reward training means you randomly reward the behavior. You vary the interval/frequency of rewards or the dog will soon figure out that every other time it performs the behavior, a reward is offered. You want to reward every third time, then every other time, then every second time, then twice in a row, etc. You want to increase the number of repetitions the dog performs, before the dog gets the reward, until you can have the dog do ten reps with no reward,etc. Don't set up a pattern. The dog will soon realize that it never knows when a reward is coming so it will try to increase the probability it earns a reward by complying every time, even when there is no food reward.
The other thing that many people fail to do is to truly proof the dog. Use sit/stay for example. They don't increase the time, distance and degree of difficulty they're requiring of the dog. Asking for a faster response before rewarding, asking the dog to sit/stay as you gradually increase your distance and adding in distractions (tossing a stick, etc.) will all make for a solid reliable sit/stay. You use the same type of intermittent reward schedule as you add these variations to sit or whatever command you're working on.
I find people are so intent on rewarding their dog for any semblance of compliance that they never push for more. Dogs are incredibly adept at figuring out that though they had been rewarded if they sat for three seconds but now the rewards aren't coming unless they sit for longer and longer periods. Don't be afraid to not reward. Dogs want to figure out what we're trying to teach them but they will take the lazy way out if we simply toss them treats any time they half way respond to a command. That is, if we've truly taught the dog the command. You can't start demanding a dog lie down for five minutes if you're just starting to teach the command.
Understanding how dogs, and in fact most animals, learn is incredibly helpful when training. My favorite introductory book is Don't Shoot the Dog by Karen Pryor. I highly recommend it to anyone wanting to train their dog, spouse or goldfish. :)