... 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 HTML, CSS, and JavaScript.
... 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).