The 12 discrete pitches within an octave are C, C-sharp/D-flat, D, D-sharp/E-flat, E, F, F-sharp/G-flat, G, G-sharp/A-flat, A, A-sharp/B-flat, and B. That pattern is the chromatic scale, and it is created by simply ascending (or descending) by half-steps and thus playing all possible pitches.
Since the pattern and collection of notes are the same no matter which is the starting pitch, the chromatic scale cannot be transposed. There is only one chromatic scale.
Melodies and harmonies that use pitches that cannot be contained by a single diatonic (major or minor) scale are often considered chromatic. Using the chromatic scale in a piece of music can make it sound exotic. Chromatic melodies can seem sinuous or elusive.
In the example of the chromatic scale below, notice that it is notated using sharps when it is ascending, but when it is descending the sharped notes are replaced with their enharmonic equivalents so that only flats are used. This is a common convention for notating chromatic scales.