These days it is extremely important for a web design company to offer more than just simple web design and building. Besides having to keep up with the latest trends and technologies a web designer’s day is full of twists and turns. Here are a few things you can expect your web company to be well versed in:
Know how to design

This one may be obvious but I mean more than just design a website. Print design and traditional design all have their own unique attributes, and some of those could be applied to certain web projects. If your web designer is experienced enough they will be able to pull from a vast knowledge base and supply you with the best possible design for your company.
Know how to use all Major Design Software

This more or less refers to the entire Adobe suite but it could be interpreted as just being up to date with the latest standards. If your web company is not comfortable working in Photoshop.. I would definitely shop around.
Know their code

Knowing HTML is definitely not enough to get by on these days, most web designers should be familiar with at least HTML, CSS and JavaScript. But that is the bare bones minimum, they really should have a strong foundation in server side language as well like PHP,ASP,Ruby or Coldfusion.
Are SEO experts

The greatest web design in the world is not worth anything if no one ever visits it. It is imperative that your designer has a strong foundation in SEO so that they can make design and development decision early on in the design process.
Are good communicators

You will need to work very closely with your web designer to create a website that is right for your business. It is important during this stage that the designer and client can communicate what they need to accomplish. Make sure your web designer listens to you needs and addresses them.
Another thing to keep in mind that it is entirely likely that your new website will need a Content Management System, will your web designer be able to teach you how to use it? Keep that in mind when choosing the right web design company.
Staying on top

Technology is alway changing, no matter if you build rockets or web sites. It is important that your web designer is building you a site that utilizes the latest technologies. After all who wants a web site that doesn’t work on mobile devices simply because your web designer was not up to date.
Summary
This is just barely a drop in the bucket of the type of information you should be able to get from your web designer at a moment’s notice. It is their job to be able to provide you with answers to your questions. Having said that don’t drop your designer if they can’t answer your questions immediately, they may need to find the answer after all their plate is already pretty full!