As mentioned in How Do Radios Work?, antennas work not by somehow increasing the power of a radiating wave, but rather by focusing the signal strength into a particular beam of radiation that has a greater signal strength.
Charts of radiation patterns usually show the intensity of a wave with respect to the angle at which the signal is being measured. For instance, the graphs in the diagram below show the intensity of a signal as measured along the z-x plane (elevation angle), and along the x-y plane (azimuth angle).
The 2D radiation pattern diagrams show a beam that gets weaker as you travel to the poles of the z-axis, and stronger near the "equator", while the intensity in the x-y plane remains constant no matter which direction you're facing. In 3 dimensions, the radiation pattern above would look something like the following:
Side note: it may seem odd here that the highest rating of amplification in decibels is 0dB, and all other values are negative. Does this mean the antenna is absorbing power instead of amplifying?
It's important to remember that the unit decibels is relative to a reference value. In this case, the diagram is trying to show where the signal is the strongest, rather than measuring the power relative to an isotropic antenna (if that were the case, we should have been using dBi instead!)
The 2D radiation pattern shown above is quite difficult to visualize without the 3D model. For this reason, the use of cartesian planes for graphing radiation patterns is far less common than using a polar coordinate system.
Polar coordinates provide a 360 degree view of the antenna from either a top view (azimuth pattern) or a side view (elevation pattern). Polar radiation pattern graphs contain a series of rings representing different amounts of amplification, and the position of a traced pattern on the graph shows the strength of the signal. Below is an image of a radiation pattern similar to the one discussed above transcribed onto a set of polar charts.
Say we wanted to find the signal strength of the antenna at an elevation angle of 15 degrees. Simply look at the point at which the blue line intersects the 15 degree mark and read off the approximate distance from the center. In this case, the antenna has a gain of roughly 1dB at that point. In the case of this graph it is likely safe to assume that we're discussing dBi here, though don't worry too much about that - ample context is usually provided in real datasheets.
A term relevant to transmission patterns is beamwidth. There are several technical definitions for this term, and usually the type of beamwidth definition a datasheet uses is explicitly specified, but the essential concept remains the same. Beamwidth is the "field of view" of your antenna as defined by a particular gain threshold.
The most commonly used definition is the Half Power Beam Width (HPBW), which is the field of view as given by the points at which the power of the radiation pattern falls to half of its peak value (i.e. if the peak is specified as 0dB, then the HPBW points are located at the regions where the beamwidth falls to -3dB).
There are other definitions too, such as the null-to-null beamwidth, which defines the angles for which the radiation pattern reduces to 0dB.
Note that beamwidth is usually defined separately for both the azimuth and elevation angles.
