Klystron can be divided into 2 types. One a Reflex Klystron and the other a Multi-cavity Klystron. Here explains about the Multi-cavity Klystrons which are used for amplifiers.
A Multi-cavity Klystron is structured by an Input Cavity, an Output Cavity and more than 1 Middle Cavity in between.
Following figure shows the basic structure of a Multi-cavity Klystron.