With positive reinforcement a rat learns to increasingly volunteer friendly behaviors

Positive Reinforcement is the core learning method. It encourages friendly and other desired behaviors in all animals, including pet rats. It's not just giving out treats to rats so they learn to like you, or to associate you with good food or other good things like play times.

Positive reinforcement can be used for thousands of training purposes, but this gallery focuses on the simple basic situation of wanting a rat to become more friendly. This type of rat might be kind of shy and holding back or hiding. You might assess that she kind of wants to be friends but doesn't know how. She's not overly terrified and definitely not aggressive.

If your rat is deeply frightened or aggressive, including that she bites and draws blood, a different method may be needed first. Start with How to Relate to Fearful or Aggressive Rats, and read more about the RAT-C method in Constructional Approach Treatment-Rats.

The set-up:

Have a set of maybe 15 food treats handy. One treat needs to be: tiny, that the rat can eat in less than 5 seconds, not too caloric (because you want to be able to give out many of them in one session), and, most important, that the rat likes. It isn't a good treat unless the rat likes it. If you want to offer a treat on a long spoon, that works too.
Special note on treats: If your rat is normally fed high-fat, high-protein, or high-sugar foods, it might be harder to find a high value treat for training. In this case a long-term adjustment back to a proper diet will be not only helpful for training, but for the rat's health and weight-maintenance. (The treat I currently use, where my rats are on a stable rat diet, is Perky's Crunchy Rice Cereal. This cereal can also be found at Whole Foods Markets in the bulk section, at least in the Bay Area, California.)
Sit quietly with the rat, away from the cage, perhaps at a small play table, where the rat feels comfortable. Give her something to hide in, to help her feel safe. If she wants to explore, so much the better.
Have the treats right there where you can offer one to the rat very quickly. If you use a spoon with, say, a dab of yogurt, have that ready. It's fine to offer the spoon to the rat to lick briefly, then pull it away with yogurt remaining on it. With this method you may need to replenish the yogurt-on-spoon no more than 1-2 times. Practice these in advance: (1) Bring the spoon (or treat to the rat for a lick or munch, and pull away, and (2) Bring the spoon to the bowl of yogurt, and scoop a bit more on. Practice so your treat delivery is smooth and swift.
Plan to deliver the treats in rapid succession if the rat offers steady or increasing good behaviors quickly. Again, practice in advance.
Baby talk-type encouragement talk is not part of positive reinforcement. For this exercise, no talking. Talking can actually get in the way of the method working.

The method of positive reinforcement in a nutshell is these steps in this exact order:

Sit quietly with the rat, watch her behavior. Watch for the moment when she does a behavior that you would like to see more of. Example: Perhaps she takes a step in your direction. Or, if she's inside a box, she pokes her head out in your direction. The behavior can be very slight, so keep your keen eyes focused so as not to lose the moment.
The moment she does that good behavior, offer the treat. Timing is everything, the treat offer needs to be immediately following the good behavior. Not 3 seconds or 5 seconds later. Right after. If the rat continues to offer the good behavior, for example she continues to walk toward you, or, she ventures a bit more outside the box, then in rapid succession offer treat after treat, to reward each new positive step. If the rat backs off or stops offering the behavior, no problem, sit back and wait.
Over time with this method, the rat's good behavior will strengthen, and the behavior “sticks”. This is Learning.

A few more notes:

If your rat stays hidden or does not volunteer any behavior you want to see more of, consider that something else needs to happen first. Maybe the table is too new and unfamiliar, and the rat needs some acclimating time, or the dog needs to be put outside, or the rat is so fearful that another approach is needed first.
With positive reinforcement, we wait for the animal to volunteer the desired behavior. That might mean that you sit there doing nothing for quite a bit of time, waiting. That's okay.
On the other hand, a little bit of luring might be helpful to get the positive reinforcement ball rolling. For example if the rat is inside a box, feeling shy, you could place a treat outside the box, to entice the rat a bit. Once she starts looking interested, and you can capture behavior you would like to see more of, shift back into positive reinforcement proper, and avoid luring. A rough suggestion is not to lure more than 3 times for a behavior.
You might be able to offer 10 treats in 1 minute. This would be a great session. Take a break and try again in 5 minutes, 10 treats for 1 minute. Limit sessions to 1-2 minutes. A suggestion is to try 2-3 (1-2-minute) sessions, then quit for the day.
Offering a treat reward is properly called “reinforcing”. Offering a reinforcement. This is because over time the desirable behavior will strengthen — be reinforced.
To practice positive reinforcement well requires a “high rate of reinforcement.” This means a rapid set of treat-deliveries, one after the other, for one specific behavior in a session. Example: The rat takes a step to you; REINFORCE. She thinks for a second and takes another step, or even leans in your direction: REINFORCE. Behavior-Reinforce, Behavior-Reinforce, Behavior-Reinforce. 1-2-3-4-5-6 treat deliveries in quick succession.
If a high rate of reinforcement is desirable, a low rate of reinforcement works against strengthening a behavior. Example: The rat takes a step toward you: TREAT. She looks in the other direction and becomes distracted, so you wait. She moves around looks at things, and then randomly takes another step in your direction: TREAT. Then she sits and scratches herself, and so you wait. This more spotty reinforcement pattern will be less helpful.
So choose a behavior you would like to see more of, based on what the rat is offering you in a particular session, and work to attain a high rate of reinforcement.
Set your rat up for success. If you are cooking a stew, the smell might be the rat's focus, instead of you. Several people watching might be a little scary. If your rat feels safe under some fleece, but you take that away, her increased fear will interfere with learning. Set up the rat for success: arrange the environment, or the conditions of the training, to make it easy for the rat to try and to succeed.
Work with behaviors the rat can offer, given her state of shyness. Example: She can take a few steps towards you, so lay your arm down 6 inches away, and then reinforce her for reaching your arm and sniffing it. Be careful to not ask her to continue to climb up to your shoulder, which would be asking too much too fast.

Summary:

Have patience to work in many short sessions.

Stay focused on watching the rat's behaviors: Spot the ones you would like to see more, and time your treat or reinforcement delivery to immediately after the behavior.

Focus on the behaviors the rat offers you, and try to provide a high rate of reinforcement.

Then watch for changes in behavior over time. A rat who is positively reinforced, will demonstrate more friendly behaviors over time.

If you aren't sure you are doing it right, or don't see increasing friendly behaviors, seek help.

There is another awesome way to practice positive reinforcement, that can actually be more efficient and successful, and that is Clicker Training. Clicker training is 100% positive reinforcement, but adds a CLICK sound that marks the desired behavior. Instead of delivering the reinforcement with blistering speed right to the rat at after the good behavior — because this can sometimes be awkward to achieve — the human CLICKS at the moment of the good behavior, and then follows up promptly with the food reward. With clicker training, the rat learns that when she hears CLICK, that behavior that she just did is what her human wants more of. And so she will offer more of the desired behavior. Clicker Training is its own method that you might choose to pursue. Read about Clicker Training Pet Rats.