There are some really talented Web Designers out there.. but, unfortunately, yours might not be one of them. The following are some red flags that might mean it's time to reassess your working relationship and seek out better talent.
1. Poor Communication, or a bad attitude are immediate red flags that your web designer might need to be let go. They aren't returning your emails or phone calls. They are avoiding deadlines and giving you the impression that your requests are unreasonable (which they may be). Regardless, when the lines of communication aren't open, it may be time to move on.
2. Cross Browser Check or Fail. If you find yourself checking their work on your web browser and saying, "Hey, this doesn't look right," and they respond,"No one uses Firefox"(or Chrome or IE or anything that isn't the browser they designed for) - then consider that they may not be the most competent person for the job. Checking cross-browser compatibility is easy, and good web designers are pros at it. Uncertain how your designer stacks up? For some easy ways to check Cross-Browser compatibility, check out 7 Fresh and Simple Ways to Test Cross Browser Compatibility.
3. CM--What? CMS or a Content Management System, is a way of dynamically cataloging content in a database so that it's easy for the owners of the website to update and maintain it themselves via an easy to use editor. WordPress, Drupal, and Movable Type are all examples of popular CMS options. Unless you want a website that is literally a static business card for your company, then chances are you'd benefit from a CMS. If your web designer isn't suggesting a CMS, or seems confused when you mention it, you should be alarmed. AshWebStudio.com has a a good primer - Do You Need a Content Management System?
4. Socially Awkward Media. Integration with social media sites like Twitter, Facebook, Linkedin, Stumble Upon, Digg, etc. are essential if you're launching any kind of online marketing campaign to promote your business/website. Your web designer should be an expert at designing with these conduits in mind. Need some guidance? For some great examples of websites with well integrated Social Media, see ReEncoded.com's Social Media Integration Examples and Tips.
5. SE-Who? SEO, or Search Engine Optimization, is absolutely essential for your business' website. If your customers can't find you, then your website might as well not exist. There are lots of good designers who aren't SEO experts, but they should at the very least be aware of how their design can impact SEO, and be able to recommend a good SEO counterpart to assist in making sure your site gets traffic. If you ask about SEO, and they say "SE-Who?" then move on. Educate yourself a little on SEO enhancements with an SEO guide for Designers.
6. It's Slow. If your hosting service is equipped to handle the traffic, your websites can be colorful and graphics-intensive, and still load relatively fast. Traffic issues aside, good web designers write clean code, and use special tools to compress images while still retaining decent quality. If your site loads slowly, consider that the design might be the culprit. For those comfortable with basic web design and Firefox extensions, you should check out Why Is My Site So Slow?
7. It's Ugly, or hard to read. Look out for the excessive use of lens flares, drop shadows, embossing, or "photoshop clouds". Look at the image to the left. Bracingly bad, right? If your site gives you the same feeling... run away! It should look good. You should really feel like it captures your business and will help you achieve your goals. In addition, good web designers are experts with text. Bad typography practices include awkward line spacing, ALL CAPS on any content longer than one sentence, or the use of too many Fonts.
8. It's got... Frames! Frames are generally considered to be a "no-no". They mess up SEO, make bookmarking specific pages difficult, and - in my opinion - are almost never necessary. If your web designer sends you a "frame-based" site, they should know better - and you should find someone else. Not sold on ditching the frames? Take a look at Jakob Nielsen's Why Frames Suck Most of the Time.
9. Sound or Music. Most users find this annoying. They will avoid your website, because they were traumatized by the exploding noise on their first visit. There are rare exceptions, but in general it's best to give users the option to "Turn On Sound" rather than having them frantically lunge across their desk to turn off speakers. Although you may have requested sound on your site, a good web designer will ask you to carefully consider how it's used and when it's played.
10. Hard to Navigate. Confusing menus and/or poor organization are a red flag that either your web designer is "not getting it", or you've really confused them. Sit down together, or schedule a call and draw up a sitemap if you haven't already done so. While this type of confusion is sometimes subjective, a good web designer will be able to make it easy for any user, including yourself, to navigate the site.
11. You aren't becoming a better manager. No one expects you, as the client, to be an expert in web design. But you may be steering your designer astray without meaning to. You may be giving them mixed signals about what you want, making changes at the last minute that cause them to miss deadlines, or not trusting their guidance when they try to educate you about why your site is set up a certain way. Keep it up long enough, and you've driven them to #1 on this list - no communication or avoidance. Think it's time to fire your Web Designer? Take a look at how you've been working with them. There may be things you can do to salvage the relationship and become better at leading (and learning from) your designer.
Recommended Reading on how web designers interact with clients :
DesignTutorials4u.com - Enough is Enough, When a Web Designer has to Fire a Client
FreelanceSwitch.com - 10 Signs you Should Drop a Client Like a Bad Habit