Ibanez Tubescreamer is one of most popular guitar pedal and its schematic is copied by many companies. Sound character is basically narrow-banded and high tone is well rounded. So, what's inside of the pedal?
You can search its schematic on Internet easily (Just search by "Tube Screamer schematic") but of course it is hard to understand how it works for most of people. So let's focus only one point of the circuit.
Here is the heart of the circuit. "JRC45580" is called Op-Amp, and as you assume, it works as amplifier. Also you see one pair of "1N914", which are Diode, it limits voltage. Typically it is used to manage current direction but they are using it for signal clipping.
When audio signal level is low, they are not working. No cuurent other words. When the signal level goes high, Diodes are activated and limit signal level. Here is the secret of the sound. Their waking up speed is not quick. It is activating guradually. That makes "Softer" clip response and make the sound more "Tube" like. As you know, people love the sound.