Quality analytics programs give you data on your visitors such as how they got to your website, how long they were on your site, which pages they visited, where they left, etc. Taking advantage of this information can dramatically increase your income. Most people who install analytics simply look at traffic volume, but you'll want to do much more.
One of the easiest ways analytics can improve your site is by making page adjustments focused on the bounce rate. The bounce rate of a page refers to the percentage of people who visit one page and then click the back button, leaving your site. A poorly designed site or page will have a very high bounce rate, whereas a well designed and engaging site will tend to have a low bounce rate. There are some exceptions though. If your website or page is designed to satisfy a visitor as quickly as possible, via a single page, then a high bounce rate isn't necessarily a bad sign. But if you need to keep visitors engaged and move them through a conversion process, you'll want to get your bounce rate down as low as possible. By using analytics you can continuously change elements of your web pages and monitor the bounce rates after making those changes. If you make a change and your bounce rate increases, you'll know you need to go back to the drawing board.
The amount of time a visitor spends on your site is similar to your bounce rate in that it's an indication of visitor engagement. Assuming you want a visitor to move through a conversion process, you generally want them to spend more time on your site rather than less. This is a site metric rather than a page metric. Like the bounce rate you should look to increase the time a visitor spends on your site by making quality improvements in both the text content and the graphic design.
Another great advantage analytics will provide is the ability to see what keywords people are searching for who arrive on your site your site. By looking at these keywords you'll find additional terms to target via SEO, and may find that certain terms are more affective than others for conversions. Some keyword phrases will be more likely to lead to a sale, whereas others will not. You can see this by looking at your analytics data, and decide which terms to target through a combination of search volume and conversion rates.
All of the elements above are related to conversion optimization, but indirectly rather than directly. A decrease in your bounce rate or an increase in the time visitors spend on your site won't necessarily increase your conversion rate, as different visitors arrive on your site for different reasons. Some actions may increase time on site but decrease your conversion rate. Because the ultimate purpose of your site is to make money, everything you do should be looked at through the lens of conversion optimization.
Regardless of what a conversion is for you, a sale, a lead, an ad click, etc., there will be a conversion path. Unless the conversion only requires a single click on arrival (in the case of an ad click), you'll have the opportunity to optimize micro steps along that conversion path. Work at improving each step or micro conversion rather than only focusing on the macro conversion. By improving each micro conversion you'll be improving your macro conversion rate.
Many web hosts provide some kind of free analytics program, but they're rarely worth looking at. The best free analytics program, and the most widely used, is Google Analytics. Google Analytics is outstanding, and not only provides tons of ways to look at your user data, but can also be integrated with Google Adsense, Adwords, and Google's Website Optimizer. Although Google Analytics if "free", there are some catches. You'll be handing your data over to Google, and you must realize that Google is not your partner (regardless of what they say), but your competition. Google is getting into more and more niches on the web, and it's highly likely that if they see your business is profitable they'll eventually get in and compete against you. However, if you're using other Google products on your site, such as Adsense, Google will have access to your user data anyway. If you're not going to be using Google products on your site at all, you should consider other paid analytics software such as Clicky.