Leonardo describes the relationship between Benedict and Beatrice thusly,
There is a kind of merry war betwixt Signior Benedict and her. They never meet but there's a skirmish of wit between them.
In the very next few lines, we see this "merry war" Leonardo is referring to. It is a war of wits as each insults the other through puns, jokes, sarcasm, and irony. What is obvious to all but the two young people is that they are in love with each other, and their outward shows of wit cover up their true feelings. The scenes in which they engage in this competition of insults, of sorts, are some of the most entertaining in the play.
So let's look at their first dialogue more closely. Beatrice begins the attack by claiming that Benedick has no need to talk because no one is paying attention to him. He returns the insult by calling her "Lady Disdain" and feigns surprise that she is still living. She answers with the claim that Disdain cannot die when it has such meat as Benedick to feed on. And Benedick vows that all women except Beatrice love him, but he loves no one. And on it goes until Benedick ends with comparing the speed of his horse to the speed of Beatrice's tongue. But Beatrice gets the last word by saying that he always ends with a "jade's trick." This is a horseman's term for an abrupt stop.
You get the idea. What is important to note, however, is that intellectually the two are matched very well. Both are proud, smart, quick-witted, funny, and not unkind. Unlike in The Taming of the Shrew, Shakespeare does not let Benedick get the upper hand in the relationship (as Petruchio did). We see this through their dialogue in Act 1 and are anxious to see how this relationship develops.