Plato's three elements of the psyche are
The appetites, which includes all our myriad desires for various pleasures, comforts, physical satisfactions, and bodily ease. There are so many of these appetites that Plato does not bother to enumerate them, but he does note that they can often be in conflict even with each other. This element of the soul is represented by the ugly black horse on the left.
The spirited, or hot-blooded, part, i.e., the part that gets angry when it perceives (for example) an injustice being done. This is the part of us that loves to face and overcome great challenges, the part that can steel itself to adversity, and that loves victory, winning, challenge, and honor. (Note that Plato's use of the term "spirited" here is not the same as "spiritual." He means "spirited" in the same sense that we speak of a high-spirited horse, for example, one with lots of energy and power.) This element of the soul is represented by the noble white horse on the right.
The mind (nous), our conscious awareness, is represented by the charioteer who is guiding (or who at least should be guiding) the horses and chariot. This is the part of us that thinks, analyzes, looks ahead, rationally weighs options, and tries to gauge what is best and truest overall.