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A semitone (British English) (also called a half step or a half tone) is the smallest musical interval used in Western music. It is the distance between two notes which are next to one another in pitch.
A whole tone means a distance of two semitones, i.e. the distance between two notes which are separated by one other note in pitch.
This is easy to see with a picture of a keyboard. The distance between two white notes that are side by side may be a whole tone (if there is a black note in between them) or a semitone (if there is no black note between).
To go from a C to a C sharp (or D flat) is a semitone.
To go from a C sharp (or D flat) to a D is a semitone.
To go from a C to a D is a tone.
An octave is divided into twelve semitones. These semitones are exactly equal in size.
- Semitone, half step, half tone, halftone, and half-tone are all variously used.
Aaron Copland, Leonard Bernstein, and others use "half tone". 
One source says that step is "chiefly US", and that half-tone is "chiefly N. Amer."
- Miller, Michael. The Complete Idiot's Guide to Music Theory, 2nd ed. [Indianapolis, IN]: Alpha, 2005. ISBN 1592574378. p. 19.
- Grout, Donald Jay, and Claude V. Palisca. A History of Western Music, 6th ed. New York: Norton, 2001. ISBN 0-393-97527-4.
- Hoppin, Richard H. Medieval Music. New York: W.W. Norton, 1978. ISBN 0-393-09090-6.