Advanced Placement (AP) courses are curricula and exams created by the College Board (who also run the university entrance SAT exam). AP classes are much more rigorous than the general course offerings (including those characterized as "honours") at American high schools....and also very much needed to beef up normal US high school diplomas if one is applying to a US uni (although IB courses will also do the job).
A very good school will have as many as 14- 20 AP courses on offer (so over three or four years, a student can take AP Algebra, AP Calculus, AP Biology, AP World History, AP English etc) which gives extra weight to a student's Grade Point Average (GPA). [For a fast and easy explanation of this mysterious calculation, go to Background Info on American Schools and Universities: You'll Need This! and scroll down to "What is a GPA?"]
"Advanced Placement" means just that -- AP courses are considered to be more advanced than normal high school courses, and are set at basic college level (meaning comparable to core courses in the first two years of university).
All AP exams are for specific subjects, and the highest score available is a 5. If a student makes a 3 or higher on an AP exam, many colleges/unis the student later attends will award credit for that subject (although in some cases, only a 5 is acceptable). Highly competitive first tier colleges and universities, however, do have their own set of requirements (see note below*).
Most American colleges/universities require students to take about two years of broad based core studies (ie, 1 math, 2 sciences, 4 English, 3 history, 1 art etc); if the uni accepts AP courses, the student can arrive at university with several credits out of the way....allowing him to skip those courses and move on into his major (the chosen course of study for which he will get his degree) sooner. As a result, he could potentially graduate sooner, meaning a lower total bill for tuition/boarding fees.
There are several advantages to taking AP courses:
they are transferable for students changing schools (a student taking the AP World History course in one school or country in the autumn can move to another school and continue on with the same course in the winter if the new school offers that course);
they give students strong and challenging course work and show university admissions offices that the student is taking the hardest classes available;
if the student's university is willing to award credit for courses taken, it can represent a significant savings on tuition fees (it is conceivable that a student could take ten or fifteen AP courses in high school, and find himself with a couple or three terms of university core studies under his belt, if some or all of the AP courses are similar enough to courses offered by the university);
The standardization allows colleges and unis to know exactly what the student's transcript means, if it has AP and/or IB Diploma courses on it.
AP courses are now also accepted by some UK universities for admission (except for some Oxford colleges).
* Worth checking: some top universities do not allow students to use AP courses for credit until their junior year, and then only if students achieved 5's on at least five of their AP tests in high school. However, some do allow students to use those credits right away.
Of course, the university only admitted that student in the first place based on the rigour of his course work (based on an IB diploma and/or a high number of AP courses!) and his excellent grades in same.