Numerous different theories have been advanced to explain why the Kaddish prayer was written in Aramaic, and not in Hebrew.
One reason given is based on the tradition that angels do not understand Aramaic. Kaddish is such a precious and valuable prayer that the angels would feel envious if they heard us recite it and they understood its meaning. This prayer was therefore composed in a language which the angels cannot understand.
The Mahzor Vitri (Rabbenu Simha of Vitri, France, d. 1105) questioned this explanation, noting that there are many other beautiful and precious prayers which we recite that were written in Hebrew, without any concern that the angels might become envious. (We might also question how angels, which are perfect beings, can experience jealousy, a human flaw. Perhaps, envy over spiritual achievements is a laudable quality, and this feeling can be experienced by angels.)
A second theory is that Kaddish is written in Aramaic as a reminder of the Babylonian exile. We emphasize to Hashem that He destroyed the Bet Ha'mikdash and drove us into a foreign land, where we spoke a foreign language, and we hope that this will lead Hashem to regret His decision and bring us back. If the angels understood this prayer, they would respond by pointing out our misdeeds, arguing that we are unworthy of redemption, and so we recite Kaddish in a language which the angels do not understand.
Another reason given is that many of the people who would attend Torah classes were simple laymen who did not understand Hebrew. Therefore, the Kaddish recited after Torah classes was written in Aramaic for their benefit, so they would understand this prayer. It seems that according to this reason, the other Kaddish recitations were modeled after the Kaddish recited after Torah classes.
The Maharam Me'Rotenberg (c. 1215-1293) suggested that we recite Kaddish in Aramaic to express our grief over the destruction of the Bet Ha'mikdash. Just as a mourner changes out of his fine garments and wears simple clothing as an expression of mourning, we, too, change the language from Hebrew, the sacred tongue, to the inferior Aramaic, as an expression of anguish. (Incidentally, some sources explain similarly why we begin the Haggadah at the Seder in Aramaic, reciting "Ha Lahma Anya." As we sit down to the Seder, we are cognizant of the fact that we are meant to celebrate this occasion in Jerusalem, with the Korban Pesach. We therefore begin the Seder in a foreign language, expressing our grief that we observe Pesach in exile.)
Rabbi Binyamin Ben Abraham (Italy, 13th century), as cited by his brother, the Shiboleh Ha'leket (Rabbi Sidkiya Ha'rofeh), suggested that the gentile authorities at a certain point forbade the Jews from reciting Kaddish. The Jews therefore began reciting it in Aramaic, so the authorities would not realize that they were reciting this prayer.
The Kolbo (anonymous work from the period of the Rishonim) offered two explanations, one assuming that Aramaic was widely known at the time of Kaddish's composition, and one assuming that it was not. If it was widely known, he writes, then Kaddish may have been written in this language specifically for the purpose of spreading its message far and wide, to demonstrate to the entire world, including the gentiles, our belief in our ultimate redemption, when Hashem's Name will be glorified throughout the world. Conversely, if Aramaic was not widely known, then perhaps it is recited in Aramaic because the angels might otherwise understand the prayer and thus prosecute against us. The Kaddish speaks of the time of the future redemption, and at that time, the righteous will be granted a greater position of stature than the angels. Since we human beings must struggle against our evil inclination to faithfully observe G-d, those who succeed and serve G-d properly deserve far more reward than the angels, who are created perfect, without sinful impulses. If the angels would understand the Kaddish, which speaks of the time of the final redemption, they might proceed to prosecute against us to ensure that the righteous would not be given a more distinguished position in the future. Kaddish is therefore recited in Aramaic, a language which the angels do not understand.