Triodes work great for many things, but as Radio Frequency (RF) amplifiers, they struggle. The problem is internal capacitance, namely between the cathode and the plate. Capacitance gets worse with frequency, so in the 1910s and 20s when radios ran on low and medium frequencies, triodes got by; but when things moved into High Frequency, something had to be done.
In the late 1920s, a fourth element was introduced, this one between the grid and the plate. This became known as the screen grid (because it physically resembled a screen mesh on a window or door), and the former "grid" became known as the control grid. (It is typical to simply refer to the control grid as the grid and the screen grid as the screen.)
The screen was connected to the plate but nearly always with a resistor in-between to drop the screen voltage; a typical screen voltage is about 70% of the plate. In action, the screen absorbed some of the electrons and created a screen current, which combined with the plate current to form a combined output current.
From a capacitance perspective, however, it changed things dramatically. If you image a triode being a glorified capacitor, with the two elements being the cathode and the plate, then a screen grid tube is really two capacitors connected in-series. There's a cathode-screen capacitance, and there's a screen-plate capacitance.
Because capacitors are Jokers in the electronics deck of cards, they work backward from what we're used to seeing elsewhere; resistance and inductance are additive: two resistors in-series combine their resistances together, same thing with coils. But capacitors are the opposite. If you want to add capacitance, you join them in-parallel. Two capacitors in-series drops the overall capacitance to less than either individual value.
Screen grid tubes could be used at higher frequencies, so they were better RF amplifiers. For this application, they swept in and displaced triodes rapidly.
I've been calling them screen-grid tubes because that was what they were commonly called when they were new; especially when it was a great marketing gimmick to get people to dump their old radios and buy new ones, or at least replace their old tubes with new ones. Over time, they became known as tetrodes: tetra = four, so four-electrodes.
And if that were all, then screen grid tubes would be supreme and this would be the last chapter. But no—there were other problems. There are always other problems, which is why we have pentodes.