Fibonacci Sequences are both natural and planned phenomena where price supports and resistances can be found. We find fibonacci sequences found in spiral galaxies, population growth, and the architecture of a nautilus shell. To the best that I can quickly explain it fibonacci sequences are a delayed-branching fractal. If anyone wants some history on this, please let me know and I will add it, but being a trading site, let's get to what you came for.

Fibonacci points, or lines rather, can drawn between the highs and lows from the most recent runs counter to the run you are experiencing. If you are in an upwards moving run, you would take the most recent low back up to the most recent high. As with trend lines, the more history that the run had, the more impact it has on the future. Additionally, the more times that the market stopped to pause at that line or deflect from it, the more it should command respect to the future.

The sequences as follows, I wrote them from memory and YES you should commit these to memory to have success in T.A.

0.000 - starting point
0.236 - minor line
0.382 - major line
0.500 - major line
0.618 - major line
0.764 - minor line
1.000 - major line
1.618 - major line
2.618 - rarely see

Let's take a look at the Standard and Poor Index (SPX)
Where are the trends? The 2008 financial crisis was our last, most obvious down or bear market. So we will examine A -> B in order to predict B -> C.
You can use to add fibonacci sequences (fibonacci retracement lines), which we will do now.
At this point we should examine it to make sure that the lines are respected by the market, this will confirm that they are properly drawn, and for the purpose of what we are doing today, confirm that they indeed do represent something significant.
So we can see that the market truly does respect these lines and they can help identify support and resistance. That top circle by the way is the .764 line, Freestockcharts does not include it as default, but anything that helps make money gets included here.

At any rate, once you have these macro lines drawn, then we can go and draw sub-fibonacci lines. Let's Do that from the most recent date, back to 2011. We will connect the 0.618 above to the 1.000 above with a new set of fib lines.
While not every move will fit exactly, you will see several deflection points as well as clusters around the sub-fib lines. When I draw the fib lines, I usually go from wick to wick on the candlesticks, but when you observe the reaction of price, you may see the body of the candlestick or the wick adhere to the line.

You can draw your lines from the candlestick body too, but you must attach the other end to a body as well. You cannot mix bodies and wicks for drawing purposes.

You can continue to place Fibonacci lines inside these as well, but as mentioned before, but they will not have the history that makes them so reliable, so it will have diminished effect. (It usually still has some effect, but diminished for each level deeper you go.)

This whole series thus far has been based on one bear run in 2008, but you can use smaller sequences too. The most recent correction, prior to this mess we are in now, was back in May 2012. You could run a fibonacci line from the low in May to most recent high in September. Guess what, it works too, and it will be different than the previous example. Try it out now. Drawing takes practice, and lots of it. Additionally, T.A. is more than just identifying fib lines, we have to use other parameters to figure out if the price will pierce it or deflect off it.

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