Many comments talk about what made google good to begin with (page rank, clean ui), but less about why it's still dominant. There are a few factors there, but the big ones seem to be company culture, scale and data.

By company culture I mean google is a very engineering focused company. They built their own everything and mainly better than anyone else. They built serialization libraries, their own server hardware and data centers, their own code repository and testing infrastructure, etc. All code must be reviewed to be used, and instead of blaming people for bugs they do post mortems and fix the underlying problems. They put everything in one giant repo so that anyone can see anyone else's code so they can fix it or reuse it. It's quite easy to just go fix something without asking your manager for permission. All this let them attract and keep great talent to keep improving things at a rate and efficiency that few other companies can keep up with. There are a few other companies that do this well, but few as well as google.

By scale and data I mean there's a rich get richer dynamic in many online businesses, including many that google is in, including search and ads. If you came up with some great idea for a search engine, could you actually take on Google? You could code it up, but could you download the entire public internet to start serving users? Could you run tests of the various tweaks to your idea are an improvement or not? There are only a handful of companies that have the scale needed to even try, and they're apparently not sure they'll succeed without some good reason (eg apple maps has a chance mainly due to the iPhone).