When you build a website, there are most likely sites like yours already. They are your competition and you need to focus on what's going to be your strategy to be better than them. Some things you can do include finding out how competing sites looked in the past and how well they're currently doing. The sites listed here will help you do these things and more.
The Internet Archive service (AKA "wayback machine") lets you see how websites looked in the past. Enter your competitors site, see how it has changed over time, and get some idea of how they operate.
Tracks the popularity of all the websites in the world by numerical rank (the lower the number, the more popular the website). You can also find out if a website's popularity is increasing or decreasing over time, pageviews per user, demographics, and more.
Very similar service to Alexa but displays somewhat different data.
NOTE: The accuracy of Alexa and Compete is highly disputed and you should be wary of the data you get from these sites.
Who Is Hosting This
Find out the hosting company of any website. By knowing the hosting company of your competitor's site you can get an idea about the attitude of the people running the site. For example, if it's a hosting company with a really poor reputation then it is not unreasonable to think that the people running the site didn't put a lot of thought into their online presence.
Find out who owns a domain name. With this information, you can get more information that could help you. For example, if the domain name is owned by a company, you can do some research on that company and find out their weakensses/flaws, and perhaps those weaknesses/flaws are present in the website as well?