Analysis begins with the rhetorical triangle--the speaker, the message, the audience. In order to analyze the piece, you first look at who is saying it, how he is saying it, what words did he choose, what is his purpose. Then you look at the message. How is it presented? Are the arguments sound or full of fallacies? Is the evidence sound, and are there lots of examples given to prove the point? Next, you look at the audience. If it's a group of women, or students, or elderly people, the message/speaker will alter language, etc. to appeal to that group. Focus on what preconceived notions the audience may have and whether or not the message would be successful with them.
In addition to the elements posted above, you should also consider allusions (lit anal), denotation and connotation (linguistic anal), and overall effect of the piece.