I would've said the Communicatve Approach has effected me the most... but the Lexical Approach covers everything in the Communicative Approach and then highlights so much more.

Some of the cutting edge course books, like Cutting Edge, and teacher training programmes, like the CELTA, have been incorporate a lot of the main ideas for years.  But teachers need to start becoming more familiar with the concepts, and employing them in addition to what they do already.

Have a proper look at it, and you'll find that it's not only based on classroom research, corpus research, and the most modern learning philosophies - it's also extremely intuitive. 

I'm starting to sound like a die-hard nothing but the Lexical Approach fan.  I'm not.  I teach pretty much the same way I always did - but there are subtle little changes in my approach that I feel make my lessons much more valuable than they used to be. 

I don't know how much this well help, as the approaches/methods mentioned above are much more detailed than I can mention in a brief response here.  And I definitely won't be showing each approach/method in it's full light.  Butit may help a bit:

 

Task based learning
Lesson staging is generally: lead in -> set up task -> Ss do task with the aim of completing the task (as opposed to 'with the aim of learning the target language') -> Ss prepare some kind of report to the whole class about the results of, or their approach to, the task -> listen to audio of native speakers doing the same task -> discuss the language involved.

The idea is that by interacting with the task (which requires a combination of skills), interacting with peers, and struggling to communicate with what they already know, Ss will learn a lot naturally.  Then by following up with a language focus afterwards, they not only learn more, but they learn items that they can see a direct need for because they just did a task in which they wanted to use it but couldn't.

Dogme
Based on the approach to cinema/movie making with the same name.  In cinema it was: back to basics - no props, no special effects, little preparation, just actors given a few minutes to play their parts in a real-life setting that matches the aims of the movie.  In language teaching it is: no lesson plan, no overhead projectors, no colourful flashcards, just teachers eliciting what students want to learn more about and using that as a springboard into the language of here and now.

Guided discovery
A technique that can be used within other approaches/methods, where the students are given a task/exercise that encourages them to analyse and think about a group of phrases/sentences to find out what is common or different between them, and to then consider a general rule to describe what's possible and what's not based on their analysis.  So it is learners discovering the rules and meaning behind language for themselves using well-designed tasks. as a guide.

Community language learning
A group of learners sit in a circle in a comfortable environment, and the teacher, who is fluent in the language being learned (L2) as well as the learners' native language (L1), walks around outside the circle.   One learner says something to another in L1, the teacher translates to L2 and drills pronunciation with the whole class, then the student who said it in L1 repeats it.  The learner he was talking to replies in L1, the teacher translates, drills, and the student who said it repeats.  And so on and so on.  Each time a student repeats the L2 version it is recorded onto cassette, and at the end of the lesson the whole conversation is played back (it will sound like the Ss simply held a whole conversation in the L2).

Communicative approach
Very general, so what can I say.  It treats meaningful interaction (both spoken and written interaction) as the most important aspect of language learning.

Total physical response
Teacher gives a command, then acts it out.  Then gives command to the students, who respond physically.  After they've become familiar with a bunch of commands, the students then give the commands to each other.  Very vaguely similar to using "Simon Says".  Another important point worth mentioning is that (especially early on) students are not forced to speak - they need only react to the language by using body language, or through drawings/ticking/crossing, etc...  Built on the idea that there's a silent period in which language learners aren't ready to produce language, that is still encouraged in Communicative and Lexical Approaches.

Lexical approach
A reinforcement of the Communicative Approach, that is also very general.  The emphasis is that the teaching of grammatical structures is much less useful than previously thought and that lexis (vocabulary, including phrases and longer expressions) is much more useful than previously thought.  It's the reason we place so much focus on teaching fixed expressions, phrasal verbs and collocations nowadays.

The silent way
A strange one this - famous for its use of Cuisennaire Rods, colour coded phonemic charts, and the teacher doing very little talking (not to be confused with TPR in which, early on, students do very little talking).  The teacher elicits a lot using the Cuisennaire Rods and pointing at the phonemic chart.  In some variations, the teacher will only model the pronunciation of a new word once (the idea being that when students get used to the fact that they won't have a second chance to hear the word, they will pay attention all the more carefully the rest of the time).

 

Hope that helps!  And do remember, it is a bit generalised.