[maven] error: package javax.servlet does not exist

The answer provided by @Matthias Herlitzius is mostly correct. Just for further clarity.

The servlet-api jar is best left up to the server to manage see here for detail

With that said, the dependency to add may vary according to your server/container. For example in Wildfly the dependency would be

<dependency>
    <groupId>org.jboss.spec.javax.servlet</groupId>
    <artifactId>jboss-servlet-api_3.1_spec</artifactId>
    <scope>provided</scope>
</dependency>

So becareful to check how your container has provided the servlet implementation.