Here's what I look for:
- Compatibility with Web sites (less of an issue these days...)
- Specific extensions (xmarks, Evernote, It's All Text)
- Desktop integration
Yes, I put stability in twice. And I meant it. Both times. That's because there's nothing that annoys me quite so much as having a browser crash mid-way through editing a page on a wiki or typing an email or doing something else in a Web-based application. Yes, GMail and other Web-based applications have come a long way in terms of auto-saving work so less work will be lost -- but I can type pretty darn quick and a crash can cost me several paragraphs, not to mention breaking train of thought. (And it's a slow train, and I prefer not to have it derailed once it picks up steam!)
As I said, these are my criteria. I know there are others. Like extensibility, or ease of writing extensions/add-ons. Or maybe you prefer a suite of applications like SeaMonkey or Opera.
Maybe the most important thing is the support community. Or developer tools.
I'd love to hear what you think the most important features are. Are there categories I've missed entirely?