A semitone, also called step or halftone, is the smallest musical interval used in western music.
On the keyboard of a piano, a semitone is the smallest change in a melody you can play. If you go a semitone up from a C4, you land on the C#4. If you go a semitone down from a C4, you land on the B3.
In digital audio workstations, the term semitone is often also called “semi” or just “st”. Therefore, if you read “+5 st”, it means adding or transposing a note by five semitone upwards.
A cent is a smaller unit, which is mainly used in digital music production. A cent is 1/100 of a semitone. The unit is often displayed at places where you tune or detune an instrument. Often you read the shortcut “ct” in software synthesizers, where you can detune a sound.
A whole tone is an interval of two semitones.
It can be confusing that a whole tone is two semitones, and therefore you move two steps in the 12 notes of the chromatic scale and not just a single one.
Be aware that not every distance between two white keys is a whole tone. The distance between E → F and B → C is only a semitone.
Usage in Modern Music
In digital music production, you will rarely use this term. Nevertheless, semitones and whole tones play an important role defining musical scales and chords. More about this later.