THE ELEMENTS OF A YAGI

THE DRIVEN ELEMENT
The driven element of a Yagi is the feed point where the feed line is attached from the transmitter to the Yagi to perform the transfer of power from the transmitter to the antenna.
A dipole driven element will be "resonant" when its electrical length is 1/2 of the wavelength of the frequency applied to its feed point.
The feed point in the picture above is on the center of the driven element.

THE DIRECTOR
The director/s is the shortest of the parasitic elements and this end of the Yagi is aimed at the receiving station. It is resonant slightly higher in frequency than the driven element, and its length will be about 5% shorter, progressively than the driven element. The director/s length/s can vary, depending upon the director spacing, the number of directors used in the antenna, the desired pattern, pattern bandwidth and element diameter. The number of directors that can be used are determined by the physical size (length) of the supporting boom needed by your design.
The director/s are used to provide the antenna with directional pattern and gain.
The amount of gain is directly proportional to the length of the antenna array and not by the number of directors used. The spacing of the directors can range from .1 wavelength to .5 wavelength or more and will depend largely upon the design specifications of the antenna.

THE REFLECTOR
The reflector is the element that is placed at the rear of the driven element (The dipole). It's resonant frequency is lower, and its length is approximately 5% longer than the driven element. It's length will vary depending on the spacing and the element diameter. The spacing of the reflector will be between .1 wavelength and .25 wavelength. It's spacing will depend upon the gain, bandwidth, F/B ratio, and sidelobe pattern requirements of the final antenna design.

BANDWIDTH AND IMPEDANCE
The impedance of an element is its value of pure resistance at the feed point plus any reactance (capacitive or inductive) that is present at that feed point. Of primary importance here is the impedance of the driven element, the point on the antenna where the transfer of rf from the feedline takes place.
Maximum energy transfer of rf at the design frequency occurs when the impedance of the feed point is equal to the impedance of the feedline. In most antenna designs, the feedline impedance will be 50 ohms, but usually the feed point impedance of the Yagi is rarely 50 ohms. In most cases it can vary from approximately 40 ohms to around 10 ohms, depending upon the number of elements, their spacing and the antenna's pattern bandwidth. If the feedline impedance does not equal the feed point impedance, the driven element cannot transfer the rf energy effectively from the transmitter, thus reflecting it back to the feedline resulting in a Standing Wave Ratio. Because of this, impedance matching devices are highly recommended for getting the best antenna performance.
The impedance bandwidth of the driven element is the range of frequencies above and below the center design frequency of the antenna that the driven element's feed point will accept maximum power (rf), from the feedline. 
The design goal is to have the reactance at the center design frequency of the Yagi = (0),,, (j + 0).
The impedance matching device will now operate at it's optimum bandwidth. Wide element spacing, large element diameter, wide pattern bandwidth, and low "Q" matching systems will all add to a wider impedance bandwidth.

ABOUT ANTENNA PATTERNS
The antenna's radiation pattern or polar plot as it is sometimes called plays a major role in the overall performance of the Yagi antenna.
The directional gain, front-to-back ratio, beamwidth, and unwanted (or wanted) sidelobes combine to form the overall radiation pattern. The antenna's radiation pattern bandwidth is the range of frequencies above and below the design frequency in which the pattern remains consistent.
 The amount of variation from the antenna's design specification goals that can be tolerated is subjective, and limits put into the design are mainly a matter of choice of the designer. "In other words.....trade offs".
Equal spaced, equal length directors may give higher gain at a particular frequency, but the bandwidth is more narrow and larger sidelobe levels are created.
Wide spacing will increase the bandwidth, but the sidelobes become large. 
By varying both the spacing and director lengths the pattern and the pattern bandwidth may be more controlled.
More directors within a given boom length won't increase the gain by any great amount, but will give you better control of the antenna's pattern over a wider range of frequencies in the band of design.
If you reduce the length of each succeeding director by a set factor (%), AND increase the spacing of each succeeding director by another factor, a very clean pattern with good pattern bandwidth can be obtained.
The TRADE OFF......will be a small loss in the optimum forward gain (10% to 15%).
In a nutshell......when you make a change to one part of the antenna, this changes the performance of another part.....all changes interact with each other and the final performance!

GAIN vs FRONT-TO-BACK RATIO
With highest forward gain design, the main lobe becomes narrower in both the elevation and azimuth planes, and a backlobe is always present. When you design "out" the backlobe, the pattern gets wider and the forward gain goes down. In some cases, the sidelobes become quite large.

A WELL FED YAGI IS A HAPPY YAGI!
There are  many ways to feed the Yagi, but they can be condensed into two main categories:
The balanced feed and unbalanced feed.

The Balanced feed system:
This may give a broader impedance bandwidth, but the main problem is that the driven element must in most cases be split in the center and insulated from the boom. Construction considerations aside, it is the better of the feed systems. Meeting the requirements of a balanced matching system is usually the main problem, but there are many methods available.
One method is to not split the driven element and use a "T" match, which can be described as two gamma matches on each side of the center of the element, fed with a 1:1 balun at the center.
The main drawback is that it's difficult to adjust.

The Unbalanced feed system:
Another method (for low impedance feed points) uses a split element insulated from the boom, and is fed with a "down-step 4:1 balun" made by combining two 1/4 wavelength sections of coaxial feedline in parallel, attaching an equal length of insulated wire to the outside of these sections, and connecting it to the center conductors at the feed point end and to the shields at the feed-line end. The impedance of this type of "balun" should be at or near the mid-point value between the feed point impedance and the feedline impedance.
For example, two 75 ohm sections paralleled will equal 37.5 ohms and will match a 25 ohm feed point to a 50 ohm feedline with a 1.0 to 1 SWR.

The most common method in use by hams today is the gamma match. It will provide an easy and sure method of matching to the feed point without any loss of bandwidth.

DESIGNING AND BUILDING A YAGI FROM SCRATCH USING THE COMPUTER!

W7RAI has a very nice DOS based Yagi design program that can be downloaded for FREE!
It will save you all that tedious math and the frustrations involved in designing a Yagi.
It will "design" a multielement Yagi for frequencies up to 999mhz. You simply download it from the link below, install it and run. You can tweak many of the variables in it to design a Yagi that will perform well for you. There is also a program within it that will design a gamma match.
Run QYUTIL.EXE in the QY4 PROGRAM from WA7RAI
for the gamma match construction details.
DOWNLOAD THE QY4 ANTENNA DESIGN PROGRAM HERE

SEE A 6 ELEMENT 2 METER SSB (144.250) DESIGN HERE 
  
Further information on antenna design and feed systems, see The Radio Amateurs Handbook, The ARRL Antenna Handbook, Dr. J.L. Lawson's Yagi Antenna Design (ARRL), or Bill Orr's Radio Engineer's Handbook, to name only a few.
Thanks to all who make Ham Radio antenna design and experimentation a fantastic sub-hobby to Amateur Radio.