Some important things to pay attention to are number of unique visits, how many of those visits are converted into sales, how visitors return to the site, and what the bounce rate is on each specific page. That last one can alert you of where potential customers get "turned off" of your product.
One of the long time players in this field is Google Analytics which allows you to set goals and track conversions. For me, Google Analytics does what I need.
Another popular tool is MixPanel and is used by a lot of startups. It has a free plan, as well as premium plans for users requiring more services. I've personally never really used it so I can't comment on it.
The most important metric to collect is conversion rate. The first thing you should be doing is to create a list of outcomes you want when people visit your website. These always begin with a verb and include things such as "Add to shopping cart," "Create an account," "Download an ebook," "Subscribe to updates," "Follow on Twitter," etc...
After you decide what your top 3 - 5 outcomes are, use the Goals feature in Google Analytics to measure your conversion rates. This will give you the best indication of how well your website is doing against your goals.
The other measurements (visitors, referral sources, bounce rate) are useful diagnostic tools but aren't a measure of how well your website is performing its mission. For example, you could have a million visitors per day but if your conversion rate is 0%, you're not doing very well. Conversely, what if your bounce rate is really high, but your conversions are at 20%? I'd take that.