The following explanation requires a working knowledge of the major scale, so you'll want to learn the major scale before continuing.
Throughout this lesson, the term "mode" (as opposed to "scale") has intentionally been used to reference the dorian. The dorian mode is actually one of seven modes derived from the major scale.
Any major scale has seven different notes (do re mi fa sol la ti, often numbered as one through seven), and for each of these notes, there is a different mode. The dorian mode is based on the second note in a major scale. Before you get confused by any further explanation, consider the illustration above.
If we were to write out the notes in the above scales, here is what we'd find: the G major scale has seven notes G A B C D E F♯. The A dorian scale has the notes A B C D E F♯ G . Notice that both scales share exactly the same notes. Which means playing a G major scale, or an A dorian scale will result in the same sound.
To illustrate this, listen to the major and dorian mp3. In this mp3 clip, a G major chord is strummed throughout, while the G major scale, and then the A dorian mode, are played. Notice that both scales sound the same - the only difference being the A dorian scale begins and ends on the note A.