Part of our function is to ensure a client's website is correctly set up so that when we do send traffic in it's direction, the user has something worthwhile to see or read. Measuring what a user does on a website has been made much easier by Google with their Analytics tools. They are the single most useful tool to a web master.
Statistics
With Google Analytics you get to see an amazing picture of what users do on your site, where they come from geographically, what site referred them to you, what page they spent the most time on and more. If you ran a campaign on a website or community forum you would be able to see how much traffic is generated from that campaign. You can see how effective your Social Media pages are in bringing you traffic. You can see if any Bloggers are talking about you and linking to your site. Once you know where most of your traffic comes from, you can focus more energy on generating more from successful avenues, or you could measure the Return on Investment (ROI) of your campaigns.
Measure
Your web-master or web designer should have spoken to you about analytics or the success rate of your website. There are too many coders and not enough business designers out there. I would rate a successful web designer as one who creates platforms for business to happen, one that makes sites that sell for the business. If your website is not making you any money, then what is it doing? What is your web-master doing? What are you paying for?
A website is the most flexible and measurable form of advertising. You can see what is working or not working and make more or less of that content on your site.
Make sure your web-master can provide you with Analytics data for your website. If you expect it to work for you, you need to know whether it ever pitched up to the office...