Some definitions of rhetoric (in no particular order)

Rhetoric refers to the ways in which “people use language to accomplish things in the world.” (Donald McCloskey, c. 1985)
Rhetoric is “the use of language as a symbolic means of inducing cooperation in beings that respond to symbols.” (Kenneth Burke, c. 1945)
“The most characteristic concern of rhetoric [is] the manipulation of men's beliefs for political ends....the basic function of rhetoric [is] the use of words by human agents to form attitudes or to induce actions in other human agents.” (Kenneth Burke)
“Something of the rhetorical motive comes to lurk in every ‘meaning,’ however purely ‘scientific its pretensions. Wherever there is persuasion, there is rhetoric. And wherever there is ‘meaning,’ there is ‘persuasion.’” (Kenneth Burke, c. 1945)
“As soon as communication tries to influence one or more persons, to orient their thinking, to excite or calm their emotions, to guide their actions, it belongs to the realm of rhetoric.” (Chaim Perelman – The Realm of Rhetoric, 1977)
Science is rhetoric all the way down. And: Science is rhetoric without remainder. (Alan Gross, 1990)
Rhetoric “is defined as the human use of symbols to communicate. This definition includes three primary dimensions: (1) humans as the creators of rhetoric; (2) symbols as the medium for rhetoric; and (3) communication as the purpose of rhetoric.” (Sonja K. Foss 2004—from our textbook, p. 4)
“Rhetoric in the most general sense may perhaps be identified with the energy inherent in communication: the emotional energy that impels the speaker to speak, the physical energy expanded in the utterance, the energy level coded in the message, and the energy experienced by the recipient in decoding the message.” (George Kennedy)
“...rhetoric is a mode of altering reality, not by the direct application of energy to objects, but by the creation of discourse which changes reality through the mediation of thought and action.” (Lloyd Bitzer)
“[Rhetoric is] that discipline which studies all of the ways in which men may influence each other's thinking and behavior through the strategic use of symbols.” (Douglas Ehninger)
“Rhetoric is an instrumental use of language. One person engages another person in an exchange of symbols to accomplish some goal. It is not communication for communication's sake. Rhetoric is communication that attempts to coordinate social action. For this reason, rhetorical communication is explicitly pragmatic. Its goal is to influence human choices on specific matters that require immediate attention.” (Gerard A. Hauser)
“...rhetoric is the process of using language to organize experience and communicate it to others. It is also the study of how people use language to organize and communicate experience. The word denotes both distinctive human activity and the "science" concerned with understanding that activity.” (C. H. Knoblauch)
“We may not be aware of all the conditions that an utterance must meet . . . there is no way we can know the conditions of success ahead of time, for they are revealed in the novel response to the utterance . . . This is the way in which rhetoric is constitutive and creative.” (Charles Bazerman, 1999)
“Rhetoric may be defined as the faculty of observing in any given case the available means of persuasion. This is not a function of any other art. Every other art can instruct or persuade about its own particular subject-matter; for instance, medicine about what is healthy and unhealthy, geometry about the properties of magnitudes, arithmetic about numbers, and the same is true of the other arts and sciences. But rhetoric we look upon as the power of observing the means of persuasion on almost any subject presented to us; and that is why we say that, in its technical character, it is not concerned with any special or definite class of subjects.

Of the modes of persuasion some belong strictly to the art of rhetoric and some do not. By the latter I mean such things as are not supplied by the speaker but are there at the outset -- witnesses, evidence given under torture, written contracts, and so on. By the former I mean such as we can ourselves construct by means of the principles of rhetoric. The one kind has merely to be used, the other has to be invented.” (Aristotle, 384-322 B.C.)
Rhetoric is “the art of winning the soul by discourse.” (Plato 427-347 B.C., who also calls rhetoric a sham art, unlike Dialectic which, Plato claims, aims to arrive at truth)
Rhetoric is that “which creates an informed appetition for the good.” (Richard Weaver c. 1950, who has been called a “militant Platonist”—any thoughts on what this means?)
“Rhetoric appears as the connective tissue peculiar to civil society and to its proper finalities, happiness and political peace hic et nunc.” (Marc Fumaroli)
“Rhetoric is the science of speaking well on civil questions for the purpose of persuading by a just and good copiousness in respect to the interactions of events and persons. Indeed, rhetoric was the term in Greece for copiousness of speaking. For among the Greeks speaking is called rhesis, and the orator, rhetor...” (Beauvais, 13th c.)
Rhetoric is “the global art that…studies…all human discourse, not just persuasion” (W. Ross Winterowd, 1972)
Rhetoric is “the study of misunderstanding and its remedies.” (I. A. Richards, 1936, who rejected the narrow prescription-based rhetoric of Aristotle in favor of a conception of rhetoric as a philosophical inquiry into how words work in discourse.)
“Rhetoric is one great art comprised of five lesser arts: inventio, dispositio, elocutio, memoria, and pronunciatio.” Rhetoric is “speech designed to persuade.” (Cicero)
“Rhetoric is the art of speaking well.” (Quintilian)
"Rhetoric is a form of reasoning about probabilities, based on assumptions people share as members of a community.” (Erika Lindemann)
“Rhetoric is the art, practice, and study of human communication.” (Andrea Lunsford)
“No art cultivated by man has suffered more in the revolutions of taste and opinion than the art of Rhetoric. There was a time when, by an undue extension of this term, it designated the whole cycle of accomplishments which prepared a man for public affairs. From that height it has descended to a level with the arts of alchemy and astrology, as holding out promises which consist in a mixed degree of impostures wherever its pretensions happened to weighty, and of trifles wherever they happened to be true” (Thomas DeQuincey, 1830)
“In short, the printed word was linked to rhetoric. While it is probably possible to invent a new rhetoric of hypermedia that will use hyperlinking not to distract the reader from the argument (as is often the case today), but rather to further convince her of an argument’s validity, the sheer existence and popularity of hyperlinking exemplifies the continuing decline of the field of rhetoric in the modern era” (Lev Manovich, 2001)
Rhetoric: “the art of using language to help people narrow their choices among specifiable, if not specified, policy options” (Hart & Daughton 2005: 2)