Any transmission line has a characteristic impedance, usually denoted Z0. If a transmission line is terminated with impedances that match Z0, a signal launched at one end will be completely absorbed at the other end, and no energy will be reflected back to the source. The voltage and/or current measured at any point along the line will be the same as any other point.
However, if a termination impedance is not matched to the transmission line, energy will be reflected back into the line, and this "reverse" signal will interfere with (add to or subtract from) the "forward" signal.
If the signal is a fixed-frequency sinewave, this interference will produce "standing waves" on the transmission line. This means that the measured voltage or measured current in the line will vary periodically with the distance from the impedance discontinuity. If the termination impedance is greater than Z0, there will be a voltage maximum at that point; if it is less, there will be a current maximum there.
The definition of "standing wave ratio" (SWR) is the ratio between the maximum voltage (or current) found at any point along the line to the minimum value found at any other point along the line. Sometimes the term VSWR is used to explicitly denote the voltage ratio. The value of this ratio is directly related to the ratio of Z0 to the termination impedance ZT. Specifically,
SWR = ZT/Z0, if ZT > Z0
SWR = Z0/ZT, if ZT < Z0
When a component or antenna is characterized with an SWR measurement, this is always specified with respect to a particular nominal transmission line impedance (usually 50Ω or 75Ω, depending on the intended application). This is just another way of stating how close the impedance of the device is to the nominal value.