HB,
Fibonacci retracement levels are derived from a simple series of numbers discovered by Leondardo Fibonacci in the 12th century.
It starts 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377 . . .
Starting with 0 and 1 each successive number is the sum of the previous two.
The ratio of each successive pair of numbers in the series soon approximates 1.618 (known as 'phi'). This approximation becomes ever more accurate as the series grows. e.g 144/89 = 1.61798
1.618 is known as the golden mean or golden ratio and it crops up frequently in nature, art, music, architecture and trading to name but a few instances.
In trading 61.8% is a common price retracement level, as are the other two. (38.2% is merely 100% minus 61.8%. I do not know why 50% is attributed to Fibonacci but I'm sure someone will).
More (slightly oversimplified) info here:
http://goldennumber.net/neophite.htm
Why this series should intrude on human market interaction nobody really knows, but Fib retracement levels certainly occur more than a random distribution would imply. Perhaps it is simply a self fulfilling prophecy, as many traders are watching these levels, perhaps not. Often worth incorporating, if only by way of a glance, into your trading IMHO.
If you look at your BARC chart as far back as the high of 633p in May 2002, you will notice that the 61.8% retracement level from this high down to the March low of 310p is 510p, almost exactly where there was a significant reversal mid September.
Regarding your M-shape question I don't think there is a specific name for this formation. However with a new leg up now it could develop into a W shape which often marks a bottom. A W shape that surpassed the previous high and Fib level at 510p would confirm the bottom and point to further upside. Conversely a failure of the support you note at 457p would point down to a new leg down to 433p or so (I think).
The M shape could also develop into a H&S if the right shoulder rallied to 490p or so and then dropped back to the neckline, which would be 457p.