The text of Kaddish, unlike virtually all other prayer texts, is written mainly in Aramaic, as opposed to Lashon Ha’kodesh (Hebrew). Although some phrases in Kaddish – such as "Yitgadal," "Yitkadash," "Yitbarach" and "Yishtabah" – are in Lashon Ha’okdesh, the majority of the text is written in Aramaic.
Tosafot (Talmud commentaries by Medieval French and German scholars) explain that the Kaddish was written in Aramaic so that the angels will not understand the recitation. Angels do not understand Aramaic, and so we recite Kaddish in this language so they do not become jealous of our recitation of these beautiful words of praises to G-d.
The Ben Ish Hai (Rav Yosef Haim of Baghdad, 1833-1909), in Od Yosef Hai (Shana Rishona, Parashat Vayehi, 1), gives a different reason (listen to audio recording for precise citation). He explains that the Kelipot (harmful spiritual forces) seek to sabotage the special spiritual effects of the Kaddish recitation, recognizing just how powerful this recitation is. Reciting Kaddish achieves "Aliyat Ha’olamot" – an elevation to the higher spiritual realms – and the Kelipot try to join this elevation in order to inflict harm. We therefore specifically recite Kaddish in a language which the Kelipot understand – Aramaic – so that they will hear the beautiful words of praise that we proclaim to Hashem, and they will be overwhelmed and subdued.
The Ben Ish Hai then proceeds to explain why the Kaddish also includes some words in "Lashon Ha’kodesh," and is not written entirely in Aramaic. He writes that when the Kelipot hear the beautiful praises in Aramaic, and then hear words in Lashon Ha’kodesh which they don’t understand, they become especially flustered – because they assume that the text which they do not understand must be many times greater than even the beautiful praises which they do understand. This combined effect of the Aramaic portions of Kaddish which the Kelipot understand, and the "fear of the unknown" triggered by the portions which they do not understand, ensure that the Kelipot will be powerless and unable to sabotage the great spiritual effects of the Kaddish recitation.