A scale is a sequence of notes in some order (i.e., ascending or descending).
The major scale in C is the sequence of the following notes: C, D, E, F, G, A, B. This scale is shown below in traditional notation.
Types of scales
The major scale in the example above is a sequence of seven notes and is called a heptatonic scale (from the Greek "hepta" for "seven"). Seven note scales are common in contemporary music, but other scales exist as well. Scales of five notes, for example, are called pentatonic scales (from the Greek "penta" for "five"). An example of a commonly used pentatonic scale is A, C, D, E, G.
Both examples above use notes from within the chromatic scale. The term chromatic scale is used to refer to a twelve tone scale where each two adjacent notes in the scale are one semitone apart (called chromatic semitone). The chromatic scale by definition splits an octave in twelve semitones. An example of this scale is C, C#, D, D#, E, F, F#, G, G#, A, A#, B. This is one scale that, other than in music theory, is generally not used in its entirety.
Most Western music will use scales built from the twelve notes on the chromatic scale. An octave, however, can be split in many other ways than in twelve semitones. One could use, for example, 24 quarter tones to produce quarter tone scales, such as the rast or sabba Arabic scales. Some Renaissance music theorists divided the octave in 19 intervals thus obtaining a 19-tone scale. 31-interval scales, 43-interval scales, and other scales have also been used throughout music.
The common C major scale (C, D, E, F, G, A, B) and the common A natural minor scale (A, B, C, D, E, F, G) contain the same notes, but sound different and so the order of notes in a scale matters. A "mode" of a scale is another scale that contains the same notes but in different order. The major scale and the natural minor scale, for example, are sometimes referred to as two modes of the same scale (specifically the Ionian and the Aeolian of the primary heptatonic scale).
Any sequence of notes can be called a scale, but some scales obviously sound better than others. Whether or not a scale sounds good depends on what notes it uses. Since each note is just a frequency, whether or not a scale sounds good depends on the frequencies of the notes in that scale.
The frequency of a note is not set in stone. A musician could tune an instrument so that it plays A with the frequency of 440 Hz. The musician could also tune the instrument so that A plays with the frequency of 450 Hz. When the musician plays the A minor scale (A, B, C, D, E, F, G) it is not important whether A = 440 Hz or A = 450 Hz. What is important is that the rest of the notes on this scale sound nicely with whatever A the musician started with. For example, the human ear tends to interpret frequencies that are integer multiples of each other (harmonics) as frequencies that sound good together. It would be appropriate then to choose A an octave higher to have the frequency of two times the frequency of the starting A. Thus, if A = 440 Hz then it would be nice if A an octave higher was A = 880 Hz. If A = 450 Hz, then A an octave higher should be A = 900 Hz. Similarly, if A = 440 Hz then an E in the same octave could be anywhere around 660 Hz, but 660 Hz has the nice property that 660 = 440 * 3 / 2. If A = 440 Hz it would be nice to choose E as exactly 660 Hz. If A = 450 Hz it would be nice to choose E as exactly 450 * 3 / 2 = 675 Hz.
The frequency of a note is not fixed and not important by itself. The relationships between the frequencies of different notes are important in determining what notes sound good together and should thus be a part of a scale.
A scale in which the frequencies of all notes are related to each other by some integer multiples as in the example above are known as just tempered scales. A chromatic scale, for example, can be just tempered, as long as some such integer relationships exist. Heptatonic and pentatonic scales that arise from such a chromatic scale would also be just tempered. Examples of just tempered chromatic scales include the Pythagorean tempered and the Helmholtz tempered.
The problem with just tempered scales is that perhaps there are no simple fractions that can split an octave in equal intervals. For example, there are no simple fractions that can split the octave in twelve equal intervals and so a just tempered chromatic scale must use intervals between each two each adjacent notes that are not the same (for example, the interval between A and A# would be different than the interval between B and C). Hence, while just tempered scales may sound good, they are difficult to transpose. The sequence A, B, C would sound different than the sequence D, E, F, even though there are supposedly two semitones between A and B and between D and E and there is supposedly one semitone between B and C and between E and F. The semitone between B and C is not the same as the semitone between E and F.
To make music easier to transpose, we can split the octave in equal intervals. Scales that split the octave in equal intervals are known as equal tempered scales. If a chromatic scale is equal tempered then each semitone would have the same size and thus transposing A, B, C to D, E, F would produce a melody that sounds similar. An equal tempered chromatic scale would make melodies easy to transpose, but it would be just an approximation of the just tempered scale. For example, if A = 440 Hz a just tempered scale would usually set E = 660 Hz. Those two frequencies would sound good together. An equal tempered chromatic scale, on the other hand, would choose E of approximately 659.26 Hz, which is not as nice.
Each scale can be just tempered and each scale can be equal tempered depending on what frequencies are chosen for each of the notes. Just tempered scales sound better but are difficult to transpose. Equal tempered scales are easier to transpose, but do not sound as nice.
Note that most contemporary instruments are manufactured to allow only equal tempered tuning and so the scale temperament is not really a choice.
There are many examples of usable scales. An index of scales available on this site is provided in the topic Scale (index).