Modern American satire relies primarily on two rhetorical strategies, irony and invective; and it is often more effective when it is target-rich. Also, rhetoricians are already familiar with some of the most salient aspects of satire, such as ethos and kairos. Ethos, or the persuasive appeal of character, is especially important considering that many satirists focus on revealing the hypocrisy of other people. Kairos, or opportune timing, is particularly significant to a genre like satire, which needs to be timely in order for it to be relevant (after all, a satirical critique invoving FDR's fire-side chats probably would not have much impact today). In addition to these important concepts, the following terms are useful for analyzing and understanding the rhetoric of satire.
(The following definitions are quoted from The Forest of Rhetoric, and the descriptions are mine)
Hyperbole - Rhetorical exaggeration. Hyperbole is often accomplished via comparisons, similes, and metaphors. It is a typical component of both irony and invective.
Metonymy - Reference to something or someone by naming one of its attributes. This concept, along with synedoche (see below), is especially significant in satire where an object of criticism represents a larger body or group. For example, when South Park mocks Tony Hayward's apologetic advertisments (see the clip here), he represents the corporation BP and especially its board of directors. Similarly, satirists will often target individual celebrities as a way of criticizing cultural values.
Synecdoche - A whole is represented by naming one of its parts (genus named for species), or vice versa (species named for genus). (For a further description see metonymy).
Other rhetorical concepts have varying levels of prevalence in modern satire. Robert Harris's analysis of eighteenth-century satire lists many of the techniques still in use today: "Certain specific literary techniques and constructions lend themselves easily to satire because they can contain a measure both of wit and humor, and of the necessary irony or satiric association; among them are exaggeration, distortion, understatement, innuendo, paronomasia, zeugma, ambiguity, what I call 'the list,' simile, metaphor, oxymoron, parable, and allegory."