The word convince supports various uses in our language.
One of the most widespread uses of the word allows to express the action of persuading and with it getting another person to start believing in something that is suggested or finally decide to carry out an action, which is also proposed to you. I am going to convince Maria to change places for lunch. Laura convinced me that this skirt was not for me even though my size fit well.
Who is able to convince someone is because he has persuasion, which is the human capacity to convince someone of something, while it is also one of the most important methodologies that exist to influence the social sphere and finally achieve the change of position, an idea, which is held by a majority.
Meanwhile, there are various ways to achieve this conviction of something, among which the following stand out: reciprocity (there is an inclination to return a favor to the other), commitment (when someone agrees to support something it is fulfilled and rarely unobserved), social proof (It is common for us to repeat those actions that we observe in others), authority (we usually trust the messages of those people who hold authority in our community), taste (when someone with whom we feel comfortable encourages us to do something, it is easy for us to accept their proposal), shortage (The perception of something as insufficient or as soon to disappear will increase its demand).
On the other hand, we use the word convince to express the pleasure or satisfaction that something or someone causes us. Juan convinces me so I'll go out with him again.
Also, from the word convince we can indicate the security we feel about something.
Then, persuade is the synonym that we use the most as an alternative to the word convince, while talk out, will be the term that is directly opposed to the one at hand, since it implies urging another respect to desist from taking any action.