“Volume Swell” is a technique used by guitar players to emulate the sound of a violin being played. What it does is, introduce the note slowly instead of realtime reproduction  in normal playing. The speed of bringing the note to fullness depends on the method used to achieve the swell.
Methods:
There are two methods you can go about to produce the sound,
1.) Using the Volume pot
After hitting the note, the volume pot is gradually turned up from 0 to 10 to produce the desired  volume level and if needed, a subtle vibrato is added.
The important thing to grasp here, is that the actual note is hit a moment before the actual sound is desired, so that the note is full and rich even before the pickups send the signal to the amp. You’ll have to be a bit careful though, because although you have to let the note ring before turning up the volume on the guitar, if you wait for too much time the note will not ring as clearly as you might want it to, depending on if you plan to sustain the note. The ideal way is to hit a note and in the same jerk of hand, turning up the volume notch with the same motion so as to get the best out of the note.

 
 
The Crying wailing signature sound of Roy Buchanan can also be achieved using this method combined with a crybaby and a nice sounding guitar.
 
2.)Using a Volume pedal
This is quite the same technique with the only difference being that instead of the volume notch, a foot pedal is used to manage the increment in sound. The output sound is much the same, though there might be a difference in the tonality depending on the signal chain in use.
 
Volume swells were made popular by blues musician Roy Buchanan who extensively used this technique combined with a crybaby wah to produce his signature tone.  Eddie Van Halen also popularized it quite a bit by releasing the song “Cathedral” which largely consists of Rapid volume swells in succession to each other.
 
Nowadays, to make it a bit easier  for beginners, some multieffects processors (the zoom “g” series for example) already have an effect in them (usually called “Slow release” or “Slow attack”) to emulate the same sound but usually offer less control than their manual counterparts.