Web counters keep track of how many people have visited your site. You put an <IMG> tag on your page which links to a 'virtual image' file on the web counter server (the HTML for this is supplied by the counter system when you create your account - you just paste it into your page source). When someone views your page, their browser asks the counter server for the 'image'. The counter system intercepts the request, updates your hit count, and generates a graphic representing the new count value, which is then displayed on your page.
Adding a counter to your home or index page lets you monitor overall site traffic, although there's a danger that people will bookmark specific pages within your site, then visit them directly without updating your home page counter. If your site has major sections then it's a good idea to put counters on the section header pages too.
Counters have been around for a long time, but the latest generation provide a valuable new feature, detailed statistics. These tell you at what times your visitors came, where they came from, which operating systems they were running and which browsers they used. The latter is especially useful, as it helps you decide which browser-specific features (such as Dynamic HTML) are worth using in your pages.
It's worth checking your ISP/webspace provider first for a hit counter, as many provide them, and an 'in-house' counter will normally give the fastest response. Alternatively, banner-exchange specialists LinkExchange are offering a new 'FastCounter' service, with statistics, at www.fastcounter.com. My current counter favourite is www.thecounter.com - it's fast, and its counter graphics are neat and carry no adverts. Its statistics are excellent; you can view them from its web site, and have a summary sent to you by email each week.