Pragmatics is the use of language, as opposed to its form (morphology and syntax) or its content (semantics). Gritchka (above) is speaking of the context and subtext of language, which is an important part of communication. However, there is more to pragmatics than just understanding the context in which something is said. Pragmatics also refers to the rules of social language, including knowing what context you need to provide to the listener, the rules that frame language interactions, and what is socially appropriate.
Conversational abilities are an important part of our pragmatic skill set. Even before we can talk we start to learn about the rules of conversation. When parents speak baby talk ('parentese' or 'motherese') to their infants, they do so in a very structured way. They use exaggerated pitch changes, to engage the infant's interest and highlight that 'these sounds that I'm making now are important. They articulate more carefully than they do with older children and other adults, to help the baby understand which sounds are important in their language. And they treat the interaction as a real conversation, keeping up a turn taking format; even if the baby doesn't respond, the pause between the parent's utterances is exactly what it would be if they were in conversation with an adult. This last aspect of parentese is an important part of training children in holding conversations, and indeed, children can take turns in a conversation and make fake words that 'sound right' before they ever speak their first word.
As we grow older we learn more rules about the form that conversation should take; questions are followed by answers, long pauses are demands for more speech on your part, and a hundred other little unspoken rules of language. All of these are part of pragmatics. It can be quite painful to talk to someone who does not recognize the turn-taking nature of conversation, who pauses too long before responding, or who fails to indicate that they are still planning to say more by interjecting an 'um' or an 'ah' into their speech.
Another important pragmatic skill is code switching (AKA social registers), the ability to switch between different communicative styles depending on whom you are talking to. You may speak very informally to your sister, in a more formal manner to your mother, and in an much more formal manner to your boss. It is an important pragmatic skill to know who you can call 'honey' and who you should not make rude jokes to. Most of us can switch between any number of 'codes' with little practice, and suffer embarrassment when we respond inappropriately (remember the time you thought it was your friend calling and you accidentally greeted your grandfather with "wazzaaaaaap!"?)
We have a number of different vocabulary markers to go along with code switching; you may use the words 'yeah', 'yep', 'yes', and 'I believe so' with exactly the same meaning but not with the same people. Code switching also takes place between situations; you may be with exactly the same people in church and then later at the beach, but you may still change your conversational style to reflect the setting you're in.
Kinesics are part of pragmatics too. The way you hold your body, where you look, and the gestures accompanying speech provide context to help the listener understand what you are trying to say and how you feel about saying it. You may feel awkward holding a perfectly normal conversation with a partner who refuses to make eye contact, because this is a sign that they are uncomfortable. You may likewise feel uncomfortable speaking to someone who won't stop staring at you. (I won't say much about body language, as we've all read a lot about it elsewhere, but rest assured that if the last few articles you read on body language were crap, that's only because most pseudo-psychologists have no interest in enumerating the unspoken rules that we all know. The real money lies in inventing new 'secrets'.)
Autism and Asperger syndrome are disorders in which pragmatic skills are particularly low. Individuals with these disorders often don't know or don't care to follow the pragmatic rules that most people take for granted. People with Autism Spectrum Disorders tend to avoid making eye contact, often don't monitor their conversational partner's interest in what they are talking about, may have trouble initiating, maintaining, and terminating conversations, may ask inappropriate questions, and have a decreased use of gestures.
Of course, teenagers also often go through an awkward stage in which they seem to loose many of their pragmatic skills, probably due to the new demands involved in acting like an adult (or almost, anyway) and dealing with members of the opposite sex, neither of which they get much training in. But that's pure speculation on my part.