The antenna for a radio is a tuned circuit that provides a "load" for the transmitter.
Just like a stereo needs to have speakers that are matched (usually 8 ohms) to the amplifier to perform properly, a radio needs a matched (tuned) load to perform properly.  Serious damage can result from operating  a transmitter with a mis-matched antenna.

The purpose of Antenna tuning is to match the load of the antenna to the transmitter.  The term SWR stands for "Standing Wave Ratio".  It refers to how much (what percent) of the radio's power is not being used by the antenna, and is left "standing" around in the coax cable, acting as a road block for the power and reflecting it back to the transmitter.  This reflected power makes the transmitter work harder than it was designed to, and creates extra heat that can destroy the power transistors in the radio.  In a perfectly tuned antenna (theoretically), there is no standing waves, and the antenna is utilizing all the power the transmitter it sending down the line.  This is ideal, because all your power is being propagated into the air and you have peak efficiency as far as the transmitter is concerned. How efficiently that power is placed, and made available to the  receiving radios, is a function of the antenna's design and placement, and not it's tuning.  However even a well designed antenna in a good location will not perform well if it is not tuned properly.

At CB frequency (27mhz), SWR should always be below 1.5:1. Above this figure, heat is starting to become a factor. By the time you reach 2:1, it is becoming critical, by 3:1, you have impending disaster, and "meltdown" is emanate!  Interference is another problem that arises after 1.5:1 SWR.  The unused power can start to cause interference in any electronic devise in proximity of the radio.  These include, but are not limited to, TVs, telephones, stereos, even the vehicle's engine management systems, and ABS circuitry.  This is not something to take lightly.  Know what your SWR is before you operate your radio.
 

Tuning

Antenna tuning is a trial and re-trial process, that if done carefully, will have no errors. Penitence is the key.

The first step is to install an SWR meter in line with your antenna, and take a couple of readings for a starting point (follow the manufacture's instructions on calibrating the unit first). Check the lowest frequency (Ch 1), and the highest (Ch 40), and compare the two readings. With a new un-tuned antenna, the 1st reading should be lower than the second.  This means the antenna is too long, and needs to be trimmed¹.  Don't be alarmed if the initial readings are very high (even over 3:1), most antennas come much longer than needed, to facilitate tuning for a wide range of applications.  Just be quick to take your readings, and un-key the radio ASAP.  Start conservatively, by trimming ½" off, then recheck the two readings.  If no noticeable change, take a 1" cut and recheck.  As you approach the correct length, the lower frequency will become acceptable first, at this point you need to start taking a reading on Ch 20 as well, because when you're done, ideally this will be the point of lowest SWR.  You should reduce your cuts to ¼" at this point, as you fine tune it.  A good well placed antenna should be capable of providing a "flat match" (reading 1:1) on Ch 20, and less that 1.5:1 on Ch 1 &  Ch 40.  If you have a mag-mount or cheap antenna, or your antenna is poorly placed, your readings will be higher.  You may not be able to achieve 1.5:1 on all channels, and will have to compromise and try to tune for the channels you use most.

For antennas that use a steel whip which require trimming as the means of tuning, you will need to have a tool capable of cutting through stainless steel. Your ordinary wire cutters are not up to the task.  There are special tools for this purpose, but they are expensive, and not worth the money for a one time trimming.  You can get by nicely with a large pair of Vise Grips that have a wire cutter built into the back of the jaw.  It will take two or more tries, tightening the  jaws down a little at a time.  Another alternative is a pair of bolt cutters.  These are cumbersome, but will cut right through the first time.

!A word of Caution!

The trimmed off piece of antenna can fly off the jaws of the cutter with enough velocity to cause very serious injury or property damage.  Always point the end you are trimming at the ground or into a container.  Also don't leave these pieces laying around where they can be picked up by a tire and cause a flat.
 
 

Notes:

¹ If you are installing a used antenna from another application, and the first reading is higher than the second,  you will not be able to tune this antenna because it is too short already and you will need to install a new one unless the SWR is already within spec.