The Tarot has a confusing and rich symbolism because it has a confusing and rich history. Not impenetrable, however; much of the fog around the symbolism can be dispelled if one bothers to study sources other than occultists with a vested interest in the mystery of it all. We'll do some dispelling further on; in the meanwhile, the most important thing to note is that modern, occult readings of the cards often have little to do with their meaning in their original context -- and that, given the modern uses of the Tarot, this is actually a good thing.
Today's Tarots have become far more interesting, expressive, and psychologically resonant today than their ancestors were. Interpretations have co-evolved with the cards over the centuries: later decks have "clarified" the pictures in accordance with their perceived meanings, the meanings in turn modified by the new pictures. Both images and interpretations have been continually reshaped, partly at random and partly in conscious or unconscious efforts to help the Tarot live up to its mythic role as a powerful occult instrument.
For example, take a look at the Rider-Waite-Smith Strength card.
We can know more about the symbolic intentions of the designer here, since he conveniently wrote many books on the subject. As with its Marseilles-deck ancestor, the card shows a woman holding the jaws of a lion, but this picture is far more elaborate.
The strangely-shaped hat of the Marseilles card has traditionally been interpreted as a symbolic lemniscate: the sideways-figure-eight representation of infinity. In the newer card, this symbol appears explicitly. Other symbols are included: a chain of roses symbolizing desire or passion, against a white robe symbolizing purity.
The mountains in the background demonstrate another kind of strength. Even here there is room for interpretation: the card is sometimes considered as showing intellect triumphing over desire, sometimes as the equal union of intellect and passion, sometimes just as a symbol of mental strength or endurance.
The twenty-two cards most often in the major arcana are: Fool, Magician, High Priestess [or La Papessa/Popess], Empress, Emperor, Hierophant [or Pope], Lovers, Chariot, Strength, Hermit, Wheel of Fortune, Justice, Hanged Man, Death, Temperance, Devil, Tower, Star, Moon, Sun, Judgement, World. Each card has its own large, complicated and disputed set of meanings.
Altogether the major arcana is said to represent the Fool's journey: a symbolic journey through life in which the Fool overcomes obstacles and gains wisdom.
There is a vast body of writing on the significance of the Tarot. The four suits are associated with the four elements: Swords with air, Wands with fire, Cups with water and Pentacles with earth. The numerology is usually thought to be significant. The Tarot is often considered to correspond to various systems such as astrology, the Kabalah, the I Ching and others.