The modern names for the musical notes, is called the "Boethian notation", due to 6th century philosopher Boethius.
It uses the letters of the alphabet: C-D-E-F-G-A-B, to denote the white notes of the piano, and C# (Db), D# (Eb), F# (Gb), G# (Ab), A# (Bb), to denote the black notes, as shown in the following image.
The black note between C and D, is one semitone up from C, called "C#" (C-sharp), or, one semitone down from D, also called "Db" (D-flat).
For practical purposes, C# and Db are considered identical, even though there are subtle differences in micro-tunings. These distinctions are relevant to non-tempered instruments, such as the violin, but are usually ignored in well-tempered instruments such as the piano and guitar.
The ancient names of the notes originated with Guido d'Arezzo, an Italian Benedictine monk of the Middle Ages.
The following table shows the correspondence between the note names in ancient and modern notation:
The "scientific notation" for note names, attaches a number to each note letter, depending on the octave it is played.
For example, the "middle C", in the middle of a piano, is called C4. The immediate white note to its right is called D4, and the immediate white note to its left is B3.
A tail piano would have eight Cs, from C1 to C8.