A Web Admin, also known as a Web Systems Engineer or similar title, is a person responsible for the systems which lie behind your favorite Web sites. Designers make the pretty pictures and CSS, programmers churn out the Java/PHP/Ruby/whatnot (or .NET if you're a poor benighted bastard). Admins make it all run. Like Scotty from Star Trek, although usually working to keep the ecommerce revenue/viral videos/porn flowing rather than dilithium crystal-produced warp energy.
Uptime! Performance! Security! Efficiency! Agility! If they're "everyone's job" then they're no ones' job.
Want to be a Web Admin? Web Admins have a system administration skillset, but are not system administrators. They have programming knowledge, but are not traditional Web developers. They need to understand their business model, but aren't suits. Their job is to focus on the systems and technologies that fall into the gaps between these worlds. In smaller shops it's a part time job, but once you get to any real size it's a whole team.