There is a lot of advertising for zinc supplements on the web, most of which seems indistinguishable from advertisements for ineffective alt-medicine cures. Are these ads a hoax, or is there any actual benefit to taking zinc supplements?
Zinc, like iron, is an element that you need. If you are a 'healthy' person, eating 'normal' (these are kind-of subjective, that's why the marks) you will get enough of these in your daily food. See for advice for instance this site. You could argue that if you take some sort of special diet, like being a vegan, you might need to consider taking supplements.
Like iron, it is possible you have too little of it, but I would only start taking supplements anyway if advised by a doctor and as a response to some sort of problem: not as a general course of action.
I do not know these ads, but I suspect they might even try to sell you them to avoid problems. If you have no problems because of too little zinc, more zinc will not help you. It does not 'stack', it will not better you in any way to take more then you need.
As @egle 's link points out, there might be some value to the thought that zinc acetate can be used to decrease the duration and severity of cold symptoms, especially cough. 1. This is not how supplements are marketed as far as I know. This would be considered 'treatment' of a problem (cold).
This graph shows there are conflicting evidence for benefits of zinc supplements.