A scale is the name for a series of notes (more correctly scale degrees) running up or down in a stepwise fashion. Any series of notes ordered like this constitutes a scale but only a few are regularly used and have common names. One is the major scale, the other is the minor scale both derived from ancient modes.
A lot of traditional music, especially Scottish and Irish is termed 'Modal' because it does not follow the conventional modern major or minor scales. It is often in the Mixolydian or Dorian mode. Modes are simply the name given to the pattern of intervals between each note in an octave i.e the ways of ordering a scale.
If you look at the tables below, and compare a G major scale (G-A-B-C-D-E-F#-G) with the Mixolydian mode, you will notice the F# is an F natural. Similarly, a D major scale (D-E-F#-G-A-B-C#-D) compared with the Dorian mode shows both the C# and F# flattened.
Modes and Scales can be traced back to Greek origins, where different different scales evolved depending on the starting note. The intervals were undoubdtedly arrived at by ear (certain intervals sound 'right' to the human ear). The scales ran down from a tonic note (rather than up which is the modern method) and maintained certain intervals between notes. Two of the scales gave rise to the modern major and modern minor scales (see table below). Each scale started on a different note and descended by characteristicly different intervals.
By the Middle Ages, the church had adopted these scales, made them ascending from a tonic note and renamed them modes. The Greek scales (and medieval modes) used only the natural notes (no sharps or flats) which are equivalent to the white notes of a piano. This gives a characteristic interval between notes in the scale (see the individual mode pages). Providing these intervals are maintained for a given mode, they can in modern music start on any note (and therefore include sharps and flats).