The answer above is inacccurate in a few details. The three entechnai pisteis are appeals from ethos and logos and by pathos. One does not appeal to the "ethos" of an audience. Intrinsic (entechne, the translation via Latin cognate as "inartistic" is misleading to those who do not know Greek) ethos means persuasion by means of portraying the character of the speaker in such a way as to dispose the audience favourably to the case. See Aristotle's Rhetoric, ps.-Cicero, ad Herennium, and Anonymous Seguarianus. Note also that appeal to biographical fact is not part of the art of rhetoric proper, and that ethos is not "identity;" it is moral character as displayed by attitudes, judgement, etc. (modesty in youth, courage in middle age, etc.)
Logos, or logic, refers to formal reasoning, not to facts, which are, in Greek "pragmata," and sharply distinguished from "logoi" as being extrinsic to the art of rhetoric (Aristotle, Rhetoric, Bk I; see also Dionysius of Halicarnassus. An example of a logical appeal would be "If all men who are dishonest in small things are dishonest in great things, and X has been dishonest in small things, then we can assume that he will be dishonest in great things as well." A discussion of whether indeed X had been dishonest in small thing would be a discussion of the "pragmata". The attempt to obtain conviction by analyzing logical relations between lesser and greater is an argument from logos.
Pathetic appeals are based on the inherent emotional predispositions of an audience, e.g. respect of sons for fathers, love of country, fear of injury, etc.
Dilts, Mervin S. and George Kennedy, eds. Two Greek Rhetorical Treatises from the Roman Empire. Leiden: Brill, 1997.
Wisse, Jakob. Ethos and Pathos from Aristotle to Cicero. Amsterdam: Hakkert, 1989.
The three appeals we use in arguments are the appeals to ethos, pathos, and logos. All three are ways we attempt to convince our audience to listen to us and believe our argument, and all three work together all the time. Some arguments, though draw on one or two appeals more heavily than the others.
When we appeal to the ethos of our audience, we appeal to their identity: their values, morals, past experience, and sense of community and history. We also attempt to build our own identity as one that our audience will believe is reliable and credible. For instance, if a candidate for mayor argues, "You should elect me mayor because I will work for the good of our helpless school children," she appeals to the ethos of a community who value their children. That argument also identifies the speaker as someone who shares those values, making that speaker more trustworthy and believable to that audience.
When we appeal to the pathos of our audience, we draw on their emotions by using expressive language and/or emotional anecdotes and examples. For instance, the same candidate mentioned above might say, "I grew up in a shack and had to work every day of my life to get where I am. Vote for me because I'm a hard worker." By telling that anecdote, the candidate makes the audience feel sorry for her, and that emotion can make the argument more believable because the audience feels sorry for her.
When we appeal to the logos of our audience, we appeal to their desire for logic. Appeals to logos include citing facts, figures, statistics, and studies. Such arguments are convincing to an audience because they are rooted in cold hard data. An example of an appeal to logos might be, "You should vote for me because crime has dropped 15% each year since I've been in office." Who can argue with that? (Note that this argument also appeals to ethos and pathos--)
To decide which appeal is right for your audience, you must know who your audience is. Every person reacts differently to different appeals; some people are easily swayed by sob stories (pathos), some want to know the facts only (logos), and others want to know why they should trust you (ethos). You wouldn't, for example, try to convince a scientist to give you a job by singing an expressive song, though that tactic might work if you wanted to convince your girlfriend to get back together with you.