Enough rambling. What is implicature? It's suggestion. Saying things without saying them. Speaking between the lines. For instance, if I say, "I ate some of the cake", I mean that there's still cake left. That's implicature. If I say, "I ate some cake", there could be some left, or there could not be. That's not implicature, that's just words. Implicature is also highly contextual. If I'm asked, "Did you eat the cake?" and I answer, "I ate some cake", suddenly the sentence I just said wasn't implicature, is. I'm saying, "I didn't eat the whole cake".
There are a number of different kinds of implicature, all explained in the link above, but I want to talk about the ones that stem from violating conversational rules, because they're the coolest—and not just because they let writers slip more information into scenes than the words themselves convey.
There are four Gricean maxims that govern conversation. A guy called Paul Grice discovered them, hence the name. The maxims are (shamelessly copy-pasted from Wikipedia because while I remember the concept, I've forgotten the terms):
1. The Maxim of Quality- Be truthful. Do not say what you believe to be false. Do not say that for which you lack adequate evidence.
The point of conversations is to convey information. If you lie, this falls apart.
2. The Maxim of Quantity- Make your contribution as informative as is required (for the current purposes of the exchange). Do not make your contribution more informative than is required.
If you ramble or over-explain, the important information (if there was any) will get lost in the TMI. Also, people might punch you if you're always providing more information than required.
3. The Maxim of Relation - Be relevant.
Tangents, not a good thing. If someone asks you about your family, talk about your family, not dinosaurs, model trains, or your fear of flying.
4. The Maxim of Manner - Be Clear. Avoid obscurity of expression. Avoid ambiguity. Be brief (avoid unnecessary prolixity). Be orderly.
If you utilize an overabundance of lexemes and morphemes to convey an elementary concept, you're doing it wrong. It's much easier to have a conversation when people know what you're saying. Ambiguity is also bad because it's so often highly confusing.
Now, we assume the cooperative principle ("Make your contribution such as it is required, at the stage at which it occurs, by the accepted purpose or direction of the talk exchange in which you are engaged."), and Grice's maxims, are always in effect, because they so very often are. This means that if you're Luna Lovegood and have a habit of saying random things when people talk to you, they're going to assume what you're saying is relevant, even when it doesn't sound that way. Implicature's going to come into effect.
Violating the Maxim of Quality - Savvy writer 1: "Do you think authors bashing authors is a good idea?" Savvy writer 2: "Of course! What could go wrong?"
Implication - Writer 2 is being sarcastic. (But both speakers must know about author-bashing scandals and the speed of information flow online for it to work.)
Violating the Maxim of Quantity - "Do you like this book?" "It's incredible! Fantastic! Unputdownable! It changed my life! So inspiring!"
Implication - This is a really good book.
Violating the Maxim of Relation - "Do you like this book?" "I barely slept last night."
Implication - Speaker 2 loved the book so much they stayed up all night reading.
Violating the Maxim of Manner - "Where's the nearest good bookstore?" "There isn't one."
Implication - Either there isn't a nearby bookstore, or the nearest bookstore sucks. Possibly not the best example I could come up with, because Speaker 1 will likely have to ask for clarification, but hopefully you get the idea.
Of course, these are only four examples. Implicatures and maxim violations are all over the place, and I'm betting you use them every day. Pay attention. Look at how they work. And then, if you're a writer, use them. Like I said above, they're great ways to slip in information through subtext. Implicatures are also good ways to convey characterization, because the way characters respond in conversations can say a lot about them. Think of Luna Lovegood. Think of the man who shouts obscenities when you say hello to him. Think of the mother who responds to "I'm hungry" with "You're looking heavier these days".
Above all, remember that everything your characters say has to be relevant, and that first person narration counts as one long conversation. "They" say not to use more words than you need to, and implicature is one of the reasons why.