This is the range of angles for which the (/help/?AntennaGain Antenna Gain) remains larger than -3dB of the peak value. A beam width of 20 degrees hence implies that the antenna radiates gain -3dB or higher over 20 degrees (main beam - 10 and main beam + 10 degrees). Sometimes the same for all angles (beam is 3 dimensional) but sometimes will have an azimuth beam width of say 60 degrees and elevation beam width of 10 degrees)

{http://www.poynting.co.za/tech_training/manual_files/image019.gif Figure1}
Figure 1 : Illustrating the 3 dB beamwidth of an 8dBi patch antenna. The beamwidth is 43°

Think of it like this: 
* Narrow Beamwidth (Dish, Helical, Yagi) for Point-to-Point links
* Wide Beamwidh (omni, patch) for local area distribution

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!! Math

For a reflector antenna it may be expressed as

HPBW = a = k l / D

where k is a factor that depends on the shape of the reflector and the method of illumination. For a typical antenna, k = 70° (1.22 if a is in radians). Thus the half power beamwidth decreases with decreasing wavelength and increasing diameter.

For example, in the case of the 2 meter antenna, the half power beamwidth at 6 GHz is approximately 1.75°. At 14 GHz, the half power beamwidth is approximately 0.75° . As an extreme example, the half power beamwidth of the Deep Space Network 64 meter antenna in Goldstone, California is only 0.04° at X-band (8.4 GHz).

The gain may be expressed directly in terms of the half power beamwidth by eliminating the factor D/l . Thus,

G = h (p k / a )2

Inserting the typical values h = 0.55 and k = 70° , one obtains

G = 27,000/ (a°)2

where a° is expressed in degrees. This is a well known engineering approximation for the gain (expressed as a numeric). It shows directly how the size of the beam automatically determines the gain. Although this relation was derived specifically for a reflector antenna with a circular beam, similar relations can be obtained for other antenna types and beam shapes. The value of the numerator will be somewhat different in each case.

For example, for a satellite antenna with a circular spot beam of diameter 1° , the gain is 27,000 or 44.3 dB. For a Ku-band downlink at 12 GHz, the required antenna diameter determined from either the gain or the half power beamwidth is 1.75 m.

A horn antenna would be used to provide full earth coverage from geostationary orbit, where the angular diameter of the earth is 17.4° . Thus, the required gain is 89.2 or 19.5 dB. Assuming an efficiency of 0.70, the horn diameter for a C-band downlink frequency of 4 GHz would be 27 cm. 

Taken From (http://www.aticourses.com/antennas_tutorial.htm ATI Space and Communications)

* (http://www.poynting.co.za/tech_training/manual.shtml WLAN Installation Manual
* (http://wireless.ictp.trieste.it/handbook/C4.pdf Antenna Basics)