First and foremost I think a hit counter should be as private as the amount of cash in your wallet. I think it's absurd to settle for a digital counter, you know the one you start at 3000 something plus 22. Why would you want others to know the amount of traffic you receive...these type of counters are out there everywhere...bravenet.com and beseen.com are 2 good places you will find these and they are free.
Second, you need a counter that can distinguish between unique hits and page views. Unique hits are the number of actual different people going to your site, and page views are the number of times these people go back and forth to your home page. For example, I go to your home page, then go to another page on your website and finally go back to the home page again. That's 1 unique and 2 page views.
A counter should also be able to report where your hits are coming from, usually called referrals. Plus being able to identify who your visitors are, time zone, country, and time of day is a bonus. When you put all this info together you get a pretty good idea who your web page is attracting.
So which one is the best? I've used HitBox, SiteStatz and SiteMeter on this web site. The best one in my opinion is HitBox, it has the most detailed reports and it's free. The reports from HitBox cannot be touched by the other two, but I think it does slow down loading of the page by a few seconds and it uses a "cookie" to gather information. So unless it can fully load then drop the cookie it does not count that hit. (this is why most counters tell you to put counter near top of page) Another drawback to HitBox, it's a little slow when you go fetch reports and data. SiteStatz, is a dynamite little counter...it's private, fast, and reports are adequate. Fetching reports is simple and quick, fastest of all three. The only drawback is the reporting of pageviews. While it will give you daily, weekly, monthly and yearly reports on uniques, pageviews can only be accounted as a running total. SiteMeter is a combination of both but does not have the strengths of either. It has everything I'm looking for, daily reporting of uniques, page views and referrals, it lacks a little when it comes to monthly and yearly reports, and they use senseless graphics where plain old numbers would work better. SiteMeter can also email your reports daily. However, Sitemeter would be the one I would use if I had to make a choice!!!
Take your pick all three are password protected, great to good reporting and free.