BrowserStack is a great tool to test website design on multiple platforms and in multiple browsers. Free plan includes 30 minutes of time.
Clickdensity is a very useful heat map/usability tool. Free to use for 1 site and 1 page with 5,000 click max.
Pingdom is an excellent service to monitor server uptime and performance. Be the first to know when your site is down. Free to use for 1 site and 20 SMS.
Piwik is a free, open-source, real-time analytics program that you can install on your own server. A great alternative to Google Analytics if you want to keep the data on your own server.
Web Developer Toolbar
The Web Developer Toolbar is a Firefox/Chrome extension that adds multiple developer tools into the browser. Another must have for any web developer.