One way to understand modes is to consider the scale of C major. This is the same as the C Ionian mode. The scale that starts on the second note of C major, uses all the same notes and ends on the D an octave higher is D Dorian. The scale that starts on E and goes up an octave using all the same notes as in C major is E Phrygian, and so on.
Each mode has its own unique step pattern. The Ionian has the familiar major scale pattern: tone, tone, semitone, tone, tone, tone, semitone. The Dorian has the pattern: tone, semitone, tone, tone, tone, semitone, tone. The Phrygian has the pattern: semitone, tone, tone, tone, semitone, tone, tone.
You might suppose that C Ionian, D Dorian, E Phrygian, and so on, would all sound pretty much the same since they're all made up from the same notes – those occurring in the scale of C major. But because the tone center and the pattern of intervals is different in each case, the character of each mode is quite distinct. D Dorian, for instance, has D as its tonal center and is only one note different (the sixth raised a half-step) from D natural minor, so it sounds very much like the natural minor scale. The Phrygian mode has a Spanish flavor to it stemming mostly from the unusual fact of having the second note of the scale only a semitone above the tonal center, so that there's a strong urge to resolve downward to the root rather than the usual upward resolution from the leading note. The Mixolydian, similarly, is only one note different than the major scale starting on the same note (the seventh note being lessened by a half-step).
The modes were used in Gregorian chant and pre-Baroque early music but are also now widely used in contemporary music. Their modern resurgence began with the release of the album Kind Of Blue by jazz-trumpet-legend Miles Davis in 1959. The musicians performing with Davis were invited to improvise on the prevailing modes rather than a given melody. Thus was 'modal jazz' spawned. A decade later Frank Zappa and other experimental rock musicians began introducing different modes into rock music.
The term modal means pertaining to modes.