Ensuring that a website is accessible is a substantial task that goes well beyond finding and using an accessibility checker. One such example is that you must make sure that your visitors are able to orient themselves no matter where they enter your site. While this is very difficult for an automated checker to measure, it is much easier when done by person.
The language you use must also be appropriate for as broad an audience as possible. As your topic becomes more complex and specialized, this task becomes harder. You can't always use basic language elements to describe complex ideas. Simplifying the language on your site then becomes yet another task that requires judgment calls on a webmaster's part. This type of judgment cannot be automated with today's technology.
Search engine optimization is also a fundamental webmaster responsibility. If something is put on the web, it is safe to assume that you want people to find it. Search engines are what people use to find information on the web. As a result, a webmaster must have at least a basic understanding of how they work. Search Engine Optimization is a broad field, and can consume a webmaster's entire day, so it is something that must be balanced carefully.
These seven duties go beyond the basics that most people are aware of, but are not an exhaustive list of what all webmasters do. This article has should have brought you from having surface knowledge of what a webmaster does to seeing the tip of the iceberg. But remember, there's still a lot left to learn. Most of the remaining tasks and duties tend to be very specific to a particular webmaster. As in most other domains, each webmaster will specialize in a different area, becoming an expert in their area of choice. This level of expertise is what will make up the tasks and duties that lie beyond the tip of the iceberg. To learn more about it, invite a webmaster you know out for lunch and ask them some questions. After all, we all like getting a free lunch and talking about ourselves :)