… have a good design sense and understand the fundamentals / design principals.
… know all the major design software including the entire Adobe Creative Suite.
… have some basic video editing skills.
… know enough about server-side languages (PHP, ASP, Ruby, Python, etc) to understand how they work, what they do, and the possibilities of their use.
… know about servers, hosting, domain registrants, DNS, etc. Setting it up, and fixing it when it breaks.
… know OS X really well (and enough Windows to get by) or know Windows really well (and enough OS X to get by) and know a huge variety of utility software that goes with.
… are good photographers.
… can color correct photos and work in RAW.
… can cut clipping paths or otherwise extract objects from photos.
… have a killer online portfolio.
… are a personable, nice people that are good with clients.
… can help clients with anything even vaguely computer-related.
… are quick to adapt to new software and new technologies.
… can train fellow employees.
… can train clients on the use of their websites.
… are good communicators.
… are team players.
… have good taste in art, music and movies.
… are up to date on social media.
… are good at logic and deduction.
… are good at user experience and user testing.
… are SEO experts.
… know about and how to handle web accessibility (and the laws surrounding it)
… understand copyright laws.
… do progressive enhancement and graceful degradation techniques.
… can debug cross-browser problems and older browser bugs.
… can bring your own client base.
… are healthy, well groomed, and wear fancy t-shirts.
… can be on-call at all times for emergencies.
… have college degrees in design-related fields.
… own very nice and expensive computers full of expensive software.
… can design for mobile devices.
… are good typographers.
Partly tongue-in-cheek of course… but not entirely. The list of things a modern web designer should know is long, and each skill feels like it could be a lifetime in itself. Good luck learning that in a 2 or 4 year program (not that that isn’t a good start).