Politeness strategies are acts of diplomacy. At their core lies the assumption that your conversation partner is presuming the worse of your intentions – and therefore you should mitigate their fears.

In politeness theory, being polite is showing awareness of a person’s face. “face” means the public self-image of a person. Each person has a need to maintain their autonomy (what linguists call “negative face”) and their social inclusion (“positive face”).

Social inclusion means being viewed in a positive light and being socially accepted. Inclusion politeness addresses this need with a display of friendliness.

Autonomy means being independent and free from pressure and imposition. Deference politeness addresses this need by showing respect, emphasizing the importance of the other’s time or concerns, and possibly including an apology for the imposition or interruption. It entails maintaining social distance.
No politeness (direct strategy): Give me a pencil.
Deference Politeness: Excuse me, sir, may I borrow your pencil?
Inclusion Politeness: Hey, brother, got a pencil?
As this example demonstrates, politeness strategies are dependent on social distance, or closeness.

When you need to make a request, you can choose one of five modes of action depending on how worried you are about threatening someone’s face:

Say Nothing
If you’re very concerned about committing a face threatening act, you may say nothing and forego your request, or hope that your need will be recognized without having to ask.
Self: (search through pockets)
Other: Here’s a pencil.

Off Record
Because a request is an imposition of the speaker on the hearer, you may opt not make an explicit request.
I wonder where I put my pencil.
Off record statements are not directly addressed to the hearer. It keeps the option open for the other to act as if he or she hasn’t heard you or hasn’t understood your speech act as a request.

Deference Politeness
I’m sorry to bother you, but can I borrow a pencil for a second?
I know you’re busy, but may I ask you if you happen to have an extra pencil I could borrow?

Using a strategy that conveys respect and distance may offend people who seek a closer relationship – they may take your behavior as a rejection. A suggested mode of action would be: with unfamiliar people, start by being polite, until you know the person better. Then gauge the situation.

Inclusion Politeness
Hey buddy, can you let me use your pencil?
Hi, how’s it going? Interesting talk, ah? Listen, could you do me a favor and lend me a pencil?

Beware: Compliments can convey friendliness, but may also seem patronizing or manipulative.

Deference and inclusion politeness strategies are more elaborated, take more effort, and are less efficient than the on record strategy. This in itself demonstrates a concern for the other’s face.

On Record
You can address the other directly to express your need. This is called “on record”. Using the imperative form is called “bald on record”.
Give me a pencil!
Lend me your pencil.

You can mitigate the request by adding ‘please’ or ‘would you’.
Using the bald, on record strategy guarantees that the other will understand your request. It is used when you don’t feel the need to maintain someone’s face.
Don’t forget to do the dishes!
One context when this form is used is in emergency situations:
Watch out!
Move out of the way!
It is also used when one speaker is in a position of power over the other (for example, in the military), and can control the other’s behavior with words.

The bald on-record communication style is the only form that satisfies Grice’s conversation maxims of quality, quanitity, relevance and manner. “Politeness is then a major source of deviation from such rational efficiency.” [1]

Preparatory Phase
One way of avoiding the risk of threatening someone’s face is to use a pre-request that provides an opportunity for the other to halt the act of request:
A: Are you busy?
B: No, what’s up?
A: Can you lend me a pencil?

An example of pre-invitation:
A: Are you doing something tomorrow evening?
B: Not really.
A: Would you like to go to a movie?

An interlocutor who wants to turn down a request, an offer, a proposal, or an invitation, can mitigate the face threatening act by showing hesitation (for example, answer with a time delay), preface the answer (well…, oh…), express doubt (I’m not sure; I don’t know), preface the answer with a token Yes (I’d love to), mention obligation (I need to do X), or appeal for understanding.

Politeness is about much more than requests. Read more on the section Other People’s Fears.

Comments:
1. I kept my examples short, but politeness strategies may utilize longer conversations.
2. Conscious planning is not necessary for typical individuals to apply these strategies. Rather, they are applied intuitively, mostly obliviously, from a young age.

Reference:
[1] Politeness: Some Universals in Language Usage. Penelope Brown and Stephen Levinson. Cambridge University Press, 1987.

Recommended Reading:
Pragmatics. George Yule. Oxford University Press, 1996.
Politeness Theory on Wikipedia
Being polite? OK, but how? Simona Petrescu.
American and British Politeness Lynne Murphy on TED Talks