In the late 20s and early 30s, the state of the art for even basic radios progressed rapidly, and tube technology struggled to keep up.
One problem was this: amplification. If you think about it, a huge amount of the effort that a radio does is signal amplification, be it RF or audio. You take this incredibly tiny signal off the antenna and build it up enough to be isolated, then built way the hell up again, then the audio-frequency signal is stripped off of that, and then this AF signal is built up until it's loud enough to drive headphones or speakers.
That's a lot of amplification.
Tube amplification works by putting high voltage on the plate. In general, the higher the voltage, the more it can amplify. So you just keep bumping up the voltage until you get enough current to not-quite-melt the plate, right?
One of the problems with current hitting the plate at high voltage is called secondary emmission. That's where the electrons strike the plate so forcefully that they dislodge electrons that in the plate, and create a cloud of free electrons around it. All those electrons hitting the plate also creates heat, which starts thermionic emmission, and either way you end up with a space charge floating around the business end of the plate.
To the cathode, that space charge appears to lower the voltage of the plate. If you have 300 volts on the place (relative to the cathode) and a 50 volt space charge, the plate now looks like it's only 250 volts. That's counter-productive.
The fix for this was yet another element, called a supressor grid, aka supressor for short. It is located between the screen and the plate, and usually placed as close to the plate as possible without causing a short. The supressor may have its own pin on the tube base and can be biased as the design engineer sees fit, but most of the time the surpressor is internally connected to the cathode; so if the cathode is ground, the surpressor is ground.
Let's assume the cathode and supressor are both ground. Here's what it looks like.
To the cathode, it's effectively invisble because they have the same potential. Likewise, there's going to be virtually no cathode-supressor current.
To the plate, it's at 100% negative voltage potential, which means any electrons that might want to boil off the plate surface are going to be jammed right back in because the supressor has a huge negative electrical charge.
The supressor is not being heated (other than by the ambient temperature inside the tube), so it's own thermionic emissions are going to be negligable, which means there's going to be almost no supressor-plate current.
The screen (remember that element?) will see it as a huge negative potential relative itself, but as the supressor has no thermionic emission, there's no current to draw from it.
So the supressor does just that: it supresses the thermionic emission in the plate.
It also does something else that's very important. It adds another internal capacitor in-series with the other elements. Now we've got the cathode-screen, screen-supressor, and supressor-plate connections; that's three capacitors in-series, and you know what happens when you chain capacitors together in-series; capacitance drops dramatically.
Five-element tubes are called pentodes.
For our purposes, this is the end of the line. There are heptodes but they are uncommon in radio use. There are tubes with more elements, but typically they tend to be special purpose tubes (e.g. magic eyes) or multiple-tubes in a single shell, like duo-diodes (6H6), twin-diode triodes (12AT6), twin triodes (12AX7), and various other combinations. And there's the pentagrid, which has five grids between the cathode and the plate; this is typically a two-triode mixer, and it combines (heterodynes) the local oscillator signal with the tuned input signal from the antenna.