Gaming an algorithm would be.... The typical example is search engine optimization. Another way is, say, we have an event happening in Ottawa. We want the hashtag for that event to trend in Ottawa so that anybody in the Ottawa area sees the hashtag on their little trending bar on Twitter. The idea might be to create a whole bunch of bots that share content using that hashtag to artificially bump up how important it is on Twitter, so that Twitter's trending algorithm forces it to be front and centre for people to see. That would be a form of gaming the algorithm.
Breaking it would be gaming it to the point where the company that created the algorithm in the first place has to revamp it entirely because it's no longer doing what it's supposed to do.