You shall make an altar on which to burn the incense (Shemos 30:1). The copper altar, upon which animals were offered, atones for the body's sins. Hence its height is three amos, the height of a human being (27:1).
The incense altar, by contrast, atones for the soul's sins. Animals, which are grossly physical, cannot atone for a spiritual entity. Therefore, incense, which produces ephemeral smoke and fragrance, is needed to purify the soul (Kli Yakar 30:1).
What are sins of the soul? The Kli Yakar maintains that the soul is sullied by the sins of the body. Alternatively, sins of the soul refer to imperfections in one's character. These shortcomings require atonement even if there are no technical violations committed by the body.
Just as one must repent from sinful actions, so must one repent from bad character traits. These include anger, hatred, jealousy, and the pursuit of money, glory and food. These sins are more severe than sinful deeds, for when one sinks into these bad traits it is difficult to separate from them. And so it says (Yeshaya 55:7), "let the wicked one forsake his way and the evil one his thoughts" (Rambam Hilchos Teshuva 7:3). The Rambam's proof text refers to the way and thoughts which require teshuva. Thoughts are the aforementioned bad character traits, and the way is the lifestyle of one who posses those traits, even if active sins are avoided.
It is for these sins of the soul that the incense altar atones. These flaws of character require smoke and fragrance, which have no hard physical substance, to achieve atonement.
The incense altar is mentioned long after the description of all the other vessels of the mishkan (Chap. 25 - 27). The intervening chapters deal with the garments of the Kohanim (Chap. 28) and their initiation (Chap. 29). A person's garments represent his character traits. The Kohen's tunic is called mido (Vayikra 6:3), a term related to middos, character traits (Gra).
"You shall make garments of sanctity for Aharon your brother, for glory (kavod) and for splendor" (Shemos 28:2). Malbim explains that the word "kavod" in this context means the soul, as it often means in sefer Tehillim. Moshe Rabbeinu himself made "clothes" for the souls of Aharon and his children by enabling them to perfect and refine their character traits.
The sins atoned for by the garments include haughtiness and evil thoughts (Arachin 16a). Haughtiness leads to anger. Both must be avoided completely (Rambam Hilchos De'os 2:3). Hatred and jealousy are also products of arrogance, which prevents a person from loving another or being happy about another's successes.
Evil thoughts include the pursuit of money, glory, and food. One ought not be mentally preoccupied with such pursuits. Rather, they should be viewed as a means to the greater end of serving Hashem (Rambam Hilchos De'os 3:1-3).
The garments of the kohanim symbolize the need to refine one's character. The term midda means measure. As a garment is fitted, or measured, so must each character trait be used in proper measure.
The initiation of the kohaim was performed by Moshe. "You shall take the anointment oil and pour it in his (Aharon's) head, and anoint him" (29:7). Although Moshe had served as the Kohein and wished to continue, he selflessly abdicated the position in favor of Aharon. He poured the oil on Aharon's beard as if it were his own (Shita Mekubetzes Krisus 5b - 25).
Moshe had no evil thoughts and no haughtiness when he initiated the kohanim. Moshe's humility, and his self-abnegation in the service of Klal Yisroel, is alluded to in the absence of his name from Parshas Tetzave (Ba'al HaTurim, 27:20). These very traits are reflected in the garments and initiation of the kohanim, which comprises the bulk of the parsha.
It is only after this character refinement that the Torah introduces the incense altar which is dedicated to atoning the sins of the soul. Alas, we no longer have the Beis Hamikdosh and its vessels. But the requirement to improve our middos remains in full effect. Indeed, it is only by such an improvement that we can merit the restoration of the Beis Hamikdash.