to speak against someone or something: to contradict, deny, or set oneself in opposition.
The word is built from "against" plus "speak," so its root idea is putting words up against someone: to contradict, to argue back, to say the opposite of what has been claimed. Most of its New Testament uses are exactly this, with opponents who dispute a teaching or object to a message.
But the word stretches past pure speech. It can mean flatly denying that something is true, as with the group who deny that there is any resurrection. It can shade into settled opposition of stance or action, as when Jesus' accusers say that claiming to be a king means opposing Caesar, or when Isaiah's God stretches out his hands all day to a stubborn, contrary people. In the mouth of a servant it narrows to talking back to a master.
It often keeps company with stronger words: in Acts the hostile crowd is "contradicting and blaspheming," and in Romans the people are "disobedient and contrary." It sits near the Greek word for open resistance (ἀντιτάσσω) and the word for refusing to obey (ἀπειθέω), but its own center of gravity stays on the spoken word: to answer back, to say no, to gainsay.