A wetsuit vest is a specific type of a wetsuit top which is essentially a sleeveless top which covers you from your shoulders to your waist. It is typically made from neoprene that is from 0.5 to 1.5 millimeters thick so you can stay warm when surfing in the summer on cloudy or windy days. The main reason why people choose a vest over a full wetsuit top is because without the sleeves you can paddle much easier and faster due to the reduced resistance.
Most people wear a wetsuit vest when in water that is between 72F-77F (22-25C). Of course you can add the vest under your regular fullsuit for extra warmth in cold water.
Recently wetsuit vests have gained popularity because of the new heated model. After the first heated wetsuit was developed users and companies figured out that is was much more practical to have a heated vest that can be used under any wetsuit and turn your normal suit into a battery powered one instead of having one expensive heated wetsuit. To find out more about the heated vest and my review of it check this article.
If you want to learn more about which wetsuit goes with which water temperature and what are the factors that influence the feeling of cold check this article.