For example, in the C major scale (white keys on a piano, starting on C), the mediant is the note E; and the mediant chord is E-minor consisting of the notes E, G, and B. Therefore, Em is the iii chord in the C major scale. Also, in the A natural minor scale (same white keys, but now starting on A), the medient is the note C; and the mediant chord is C (or C-major) consisting of the notes C, E, and G. Therefore, C is the III chord in the A (natural) minor scale.
"Mediant" also refers to a relationship of musical keys. For example, relative to the key of A (natural) minor, the key of C major is the mediant, and often serves as a mid-way point between I and V (hence the name). Tonicization or modulation to the mediant is quite common in pieces written in the minor mode, and usually serves as the second theme group in sonata forms, since it is very easy to tonicize III in minor (no need to alternate notes). Tonicization of III in major is quite rare in classical harmony, compared with, say, modulation to the V in major, but mediant tonicization in major is an important feature of late romantic music.