Tube saturation or Tube Amp in Guitarists world is a type of audio saturation where the audio is saturated using the circuits within the tubes. The tube saturation gives the certain type of analog warmth to the audio signal. Vacuum tubes are used in popular hardware units and their software VST emulations to create color into the incoming audio signal.
To be able to work successfully of the tube saturation you need minimum three elements, cathode, anode and the control grid. A tube flows the current from cathode to anode, the cathode is able to heat the current as the flow of electrons with the help of heating filament placed in the valve. How much current will flow through the grid can be controlled using voltage.
You can find triode, tetrode where the triode has three cathodes, anode and control grid, the tetrode has four. This all depends on what you are trying to achieve with the setup. This setup can be found in various popular analog hardware units like the Teletronix LA2A, Tube-Tech Eq with the Passive equalizer, tube-based push-pull amplifier.
This is the reason you like how the old records have that magical sound to it. You can use VST emulations for the box setup all the producers and some engineers have in the modern world. Various VST developers have a lot of tube saturation emulation plugins available for you to choose from.
Developers live UAD, WAVES, SOFT TUBE and many more makes premium quality emulation VST. There are some nice sounding free VST plugins are also available on the internet.
Other types of saturation include Transformers, transistors, tapes, where the audio is processed through a tape machine which adds analog warmth to the audio signal. You can read the full tutorial on Tape Saturation to understand the working of the tape machine.