Drashah, or droshah, is usually taken as synonymous with "sermon," (as in "couldn't make heads or tails of the rabbi's...") This isn't quite right. The Oxford English Dictionary defines sermon as "a discourse on a serious subject, containing (especially moral) instruction or exhortation. Also, a long or tedious admonitory lecture or harangue."
A drashah should be rather different. The word comes from "lidrosh," which means to "seek" or "inquire." In this week's parashah, Rebecca, anguished by the sensation of having two babies slugging it out in her womb, goes "lidrosh et Hashem," to "ask God" what's going on (Genesis 25:22).
Solomon sets his heart "lidrosh," to "seek out" wisdom (Ecclesiastes. 1:13). The prophet Amos implores his listeners, on God's behalf, "dirshunei v'heyu," "seek Me and live" (Amos 5:4).
Each of these verses speaks of searching into something which is opaque and mysterious. Through the enquiry it becomes clearer. So, too, a drashah is an investigation into some aspect of the Torah which is currently enigmatic or obscure, with the aim of revealing and clarifying its meaning. This is not the same as a harangue, with a point obvious to everyone from the outset.
Drashah gives its name to midrash, the unique mode of Jewish biblical interpretation in which we probe and investigate the silences, lacunae and apparent contradictions of the text in order to enlarge on the meaning of the Torah.