console.log() just takes whatever you pass to it and writes it to a console's log window. If you pass in an array, you'll be able to inspect the array's contents. Pass in an object, you can examine the object's attributes/methods. pass in a string, it'll log the string. Basically it's "document.write" but can intelligently take apart its arguments and write them out elsewhere.
It's useful to outputting occasional debugging information, but not particularly useful if you have a massive amount of debugging output.
To watch as a script's executing, you'd use a debugger instead, which allows you step through the code line-by-line. console.log's used when you need to display what some variable's contents were for later inspection, but do not want to interrupt execution.