Gain is a measure of the power amplification factor of the antenna. In proper Yagi beam designs, gain is primarily a function of boom length. At VHF and UHF frequencies, very high gain can be achieved because booms can be physically long with more elements. This has the effect of increasing the transmitting and receiving power of the antenna.
And it is a measure of the signal (power) radiated or received from the front of the antenna to signals from the back of the antenna.
The Math : ​Gain=Effiency X Directivity

Radiation Patter is the result of the designer's skill in achieving good forward gain and front to back ratio. At Cushcraft, we use these skills with computer aided design to combine the right boom and element lengths with proper spacing between the elements to achieve clean radiation patterns. Think of the radiation pattern as similar to a flashlight beam. A narrow beam focuses for greater distance, while a wide beam will not reach as far. Long boom antennas will have narrower patterns which may be focused more accurately to receive and transmit distant signals.

VSWR is voltage standing wave ratio. You will find it listed as SWR in the specification charts. When properly installed, the antennas in this catalog should show an SWR of 1.2:1 or less at the center of their operating frequency range. The 2:1 SWR bandwidth is the range over which the antenna will operate without exceeding 2:1 SWR, allowing your transceiver to operate at its full power output. Mt models may be tuned or optimized for your favorite operating range CW, phone or in many cases center band.

fifty Ohms is the value RF engineers have agreed to design for in applications requiring RF connection between equipment. In most radio communications, antenna feed point impedances between 25 and 100 Ohms are acceptable. Cushcraft engineers use various techniques depending on the application to achieve the best impedance match of 50 Ohms between the antenna and connecting cable from your radio. 

Bandwidth is the frequency range over which the antenna provides optimum performance. Specific SWR bandwidths are shown for each model. The antennas in this catalog are designed so that the best gain and SWR are achieved over the same bandwidth.

Polarization refers to the plane of the antenna elements in relation to the earth, either horizontal or vertical. Cross polarization such as trying to contact a vertically polarized mobile station antenna with a horizontally polarized beam will reduce the efficiency of the link and shorten the range of communications. This theory also applies to fixed or base station antennas. Most antennas used for FM communications are mounted with their elements in the vertical plane. For OSCAR satellite communications, antennas are specially designed to give circular polarization to compensate for satellite rotation. At HF frequencies below 30 MHz polarization is less of a factor because radio signals are subject to change as they reflect from the ionosphere back to earth. 

Radiation angle is used to describe the angle which the signal radiates to or from the antenna in relation to the earth's horizon. It is applied primarily to vertical antennas. Angles of radiation as close to the horizon as possible are preferred for good communications. At HF frequencies, this is accomplished by using half wave antennas, like the R5 and R7. At VHF and UHF frequencies, multiple vertical elements, one above the other, are used to lower the radiation angle and provide gain over a quarter wave vertical.

All antennas have elements. The basic element is a half wave dipole. Yagi beam antennas use more than one element to achieve a desired gain and pattern. Vertical antennas are also a configuration of the half wave dipole. Many verticals are one quarter wavelength long, which requires some type of ground radial system for proper impedance matching. Designs like the Ringo Rangers and UHF mobile antennas use multiple half wave elements for greater range.

Beams can be described as high performance antennas. They have a center support boom and multiple elements. Beams incorporate many of the techniques described above to improve both the transmission and reception of ham radio signals. Beams are often called Yagi's after their inventors, H. Yagi and S. Uda. At VHF and UHF frequencies beams are usually designed to give maximum performance on one band. They are usually called monoband beams. Space and size may be a problem at HF frequencies so engineers have developed multiband trapped beams. Antennas like the A3S and A4S operate on 10, 15, and 20 meters. With the addition of specially designed add-on kits they will also operate on 40 meters. The tri-band beam requires less space and a smaller support structure than separate monoband beams.