In this dialogue, written c. 380 BC, Plato intends to show us the worthiness of dialectic teaching over the rival sophistic method of eristics. Plato shows Socrates conversing with a young Crito, telling him of a dialectic argument he had with a group of young disciples and two traveling sophists, Euthydemus and Dionysodorus, who are trying to drum up some business at the Lyceum.
The eristic method demonstrated by the two brothers, Euthydemus and Dionysodorus, boils down to this. The student asserts something. By means of spurious either/or questioning, they force the student to admit the opposite of their assertion. The student then asserts the opposite of their original assertion, and the process repeats. The point is that no matter what the assertion is, the sophist will use unsophisticated, pardon the expression, logical tricks to dispute it. In this way, these sophists assert, they impart the youth with virtue and wisdom.
Bullshit, says Socrates. In many other of the shorter dialogues, Plato shows us the difficulty of even defining the term 'virtue' so we know Socrates is going to give it to the sophists with both barrels. Plato sharpens his quill and goes to work.
In a series of demonstrations, the two styles of teaching are used to try to educate poor Cleinias, one of the disciples. The sophists use their eristic methods and leave the boy humiliated and depressed. Their use of either/or argumentation seems like the Monty Python sketch, "Argument Clinic." Every which way Cleinias turns, the sophists tear him down. It's a bit comic. Contrasting with this, Socrates engages the young man in dialectic, and in a short span of time, Cleinias displays an impressive grasp on the lessons taught. The youth seems to glow with the flame of wisdom kindled within him. In short, the eristic style destroys and does not create. Teacher and student are made ridiculous. On the other hand, the friendlier dialectic approach leads to true insight. The teacher and the student are ennobled.
Socrates demonstrates that there teaching has no moral value, so they cannot assert to teach virtue. Eventually, by following the sophists' line and their style of argumentation, Socrates leads them to admit that error, ignorance, speaking falsely, and so forth are impossible. Since ignorance is not possible, then what are they doing trying to teach? If you deny that ignorance is possible, teaching is also impossible or at least meaningless.
Here, the sophists are in trouble, so they change course and try to minimize their losses. They change their story to say that they intend to teach their eristic method, and that the method itself has merit and leads to virtue and wisdom. Socrates then demonstrates with another student, Ctesippus, that their method is easily mastered, and that as teachers of method, they had better keep their day jobs.
This is not the only Platonic dialogue that attacks the sophists, but it is one of the more entertaining. Socrates's humanity and humor really shine.
Sources:
"Two Kinds of Paideia in Plato's Euthydemus," by Rosamond Kent Sprague (http://www.bu.edu/wcp/Papers/Anci/AnciSpra.htm>
and Benjamin Jowett translation of Euthydemus