The word archetype is used in conjunction with Tarot to describe how the cards fit into a universal symbology. Carl Jung developed the idea of the archetype as a symbol which all human beings understand at birth.
My dictionary defines it as follows:

(In Jungian psychology) an unconscious idea, pattern of thought, image etc., inherited from the ancestors of the race and universally present in individual psyches.".
Some of the more common archetypes are "the shadow" which represents things we fear or wish to deny in ourselves; "The animus/anima which is the internal representation of the opposite sex (anima -female within a male, animus - male within a female); "the wise old man", "the great mother", "the trickster", and "the shaman".
Per Jung, every culture, society and individual recognizes these archetypical figures. They are in our myths, our history and our legends. They have the ability to elicit strong emotional reactions.

Tarot relies heavily on symbolism and pairing it with a system like the archetypes was a natural.
Do you need to understand the archetypes and Jungian psychology to read Tarot?
No you don't, but learning this material will give you additional insights into the interpretation and meaning of the cards.
I recommend you first master the basic meanings of the cards. If you have a background in psychology and are familiar with Jung, by all means use this knowledge to add depth to your interpretations, but if you are not familiar with it, don't fret, there are readers who know nothing about archetypes and do quite well.
The same can be said for Kabala, Astrology, Numerology and other systems which are often linked with Tarot.
Learning them adds depth, but not knowing them will not prevent you from being a competent reader. Different Tarot authors assign different archetypes to the various cards.
I have seen the Magician described as an aspect of the wise old man, an aspect of the trickster and as an aspect of the shaman. Obviously any worthwhile discussion of this subject is beyond the scope of this course. I am still learning about this subject myself. I just wanted to mention it so that when you hear the word archetype discussed in relation to Tarot, you have some idea of what is being talked about.