Quarter notes (1/4 notes) get their name from the fact that it takes 4 of them to make up a bar of 4/4 time. Hence the bar of 4/4 is divided into quarters, each 1/4 note represents 1 quarter of the bar, or one beat.

4/4 time (otherwise known as C "common time") is the most common time signature, therefore the numeric names for note values are derived from this time signature. The traditional European name for a 1/4 note (the American name) is "Crotchet".

Although the numeric names (quarter note, eighth note, sixteenth note etc) may be more popular in contemporary music, in a sense they are an inaccurate description of the note value. Why? Well, if you were to play quarter notes in a bar of 5/4, which has 5 beats in every bar, it wouldn't be representing 1 1/4 of the bar, it would be representing 1 1/5th of the bar, except you wouldn't suddenly call it a 1/5th note.

Quarter notes usually represent the pulse of the bar, and other subdivisions like 8ths and 16ths are usually subdivisions of this pulse. Here is a bar with 4 x 1/4 notes in it..