Persuade means to convince somebody of something. For example, if I try to get you to join the hockey team, I am persuading you to join the team. Persuading is used in persuasive writing, where you are trying to give your opinion or convince the audience of something. Persuading is also when you want somebody to believe you or agree with what you state.
You can persuade somebody in writing, by using logos, ethos, or pathos. Logos are using logical reasoning to explain it. Ethos are specifically targeted at the audience to induce memories or belief. Pathos is convincing the reader that you are trustworthy.