Quadrangle rule
These are standard rules that apply to traffic above 3000ft to lower the risk of collision.
A way to visualise this is road traffic; without this rule, you would pull up at a junction without traffic lights, everyone would be heading in different directions, all being very carefull not to collide (being on the same junction). The quadrangle converts this single junction into a set of bridges at the junction, whereby the direction you're going in decides wich bridge you go on. The bridges keep you seperated by a fixed height vertically so you can cruise through without worrying about colliding with traffic from all directions (or at least not as much)
It basically says that depending on which heading you are on, you should select a certain flight level. This rule only applies below 24000ft, and you should use your discretion in selecting an appropriate flight level.
If you are on a heading of 0-89 degrees, you should select a cruising height beginning with an odd number.
e.g. I am on a heading of 40 degrees at a cruising altitude of 12,000 feet, I should either ascend to 13,000 or descend to 11,000.
Likewise, on headings of 90-179 degrees select an altitude of an odd-thousand plus 500ft.
e.g. I am on a heading of 100 degrees at 12,000ft, so ascend to 13,500ft or descend to 11,500ft.
For 180-269 degrees, select an even-thousand cruising level.
If you are on a heading of 200 degrees at 12,000 feet, relax and have a cup of tea.
For 270-359 degrees, select a cruise level of even-thousands of feet plus 500.
If you are at 12,000 feet on a heading of 350 degrees, either ascend 500 feet or descend to 10,500.
Semicircular Rule
This is to be used above flight level 240 (24,000 ft). It is a simplified version, if you are flying a heading of 0-179, you must select an odd flight level (25,000ft, 27,000ft, 29,000ft etc), when flying a heading of 180-359 degrees you must select an even flight heading (26,000ft, 28,000ft, 30,000ft).
Both of these rules are simply there to make sure that at any given height above 3000ft, you will not find someone at the same height as you coming in the opposite direction!
Just to clarify (as this confused me immensly at first!), you select your cruising height based on your heading, not the other way around.
Hope that helps.
Dave
(Please excuse the lack of distinction between height and altitude)
edited: I started this post at about 10.30am and got distracted, only just finished, apologies for overlap