Popular pages: By looking at your logs you can tell what pages are the most popular. This way you can examine the content to see why this page gets more hits than the other pages. Overall, this will help you better shape and design your website.
Where your visitors come from: It would be good to know where your visitors are coming from so you can keep track of it for marketing purposes. If you know where the traffic is coming from mostly, you can gear your promotions in this area where marketing will do the most good.
The least looked at pages: By examining the least visited page(s) and comparing the content to the most visited page will give you a guide to follow to determine what changes you may need to make to boost your hits on this or these pages.
Operating systems: If you know what operating system and what browser they are using most often, you can gear future products (if you sell products) to that type of operating system and browser. This way you can satisfy and gain more customers. This way you can sell more products raising your profit margin too boot.
Search engine spiders: Logs can also let you know when search engine spiders visit your site. This way you’ll have a time frame so if you wanted to update your site you will know when to do it so your site can be indexed appropriately with the new changes.