VSWR [pronounced by many as 'viz-rar'] is the Voltage Standing Wave Ratio in an RF system. I have commonly seen this measurement called S-W-R in the HAM community, which is not "wrong" but it is like saying dB instead of dBm when talking about RF power measurements. When you say SWR you could be talking about the ratio of standing waves in a kiddy pool or even the ocean. Luckely no one actually pays attention to the waves in the pool, and all HAMs know what you mean by SWR.
I want to take a moment to talk about VSWR and its significance to the community.
First we need to start with Impedance. Impedance is the opposition to current flow in a system and is measured in Ohms. If you have wondered why co-ax cable is spec'ed at 50ohms, it is to maintain the same network impedance. Your transceiver is expecting a 50ohm network. 50ohm is set as the standard.. except for 75ohm networks like your cable tv and most satellite equipment. When your network is all balanced you achieve a near perfect energy transfer. In the real world this is never achieved mostly due to antennas, but we will get to that in a minute. If you have ever tried to connect your radio to an antenna using RG-11 (cable tv coax) from home depot you would have noticed desegregated transmissions, this is because RG-11 is build for a 75ohm network. You can imagine your coax as a network of inductors in series and capacitors in parallel to ground. The inductance changes with the length of the cable and the capacitance changes with the space between the center conductor and the shield. If the coax is installed properly, and the connectors have the proper impedance characteristics as well, then the transmission line will be very close to 50ohms.
Antennas are where the impedance mostly gets fooled with. You will see that antennas are marketed for certain frequency bands, this is because they have an impedance of close to 50ohms at those bands. I have little idea how the wizards that do antenna design are able to make that magic, but there is definitely an art to it. You may be able to receive a wide range of frequencies from a given antenna, but if you transmit with a frequency that is outside of the band the antenna was designed for then the impedance would be to high and you could blow up the amplifier on your transceiver.
What do the numbers mean? In a perfect world you would have a VSWR of 1:1, meaning the network is perfect and all the energy you transmit to the antenna would be radiated from the antenna. Unfortunately we live in the real world and you will see values more along the lines of 1.4:1 to 2:1 meaning that you are missing out on 2.8% to 11% getting to your antenna.
Here is a video of standing waves in a pool of water
here is a video of standing waves on a plate
and here is a video on RF standing waves