Droptor is one of the easiest ways to monitor the health of your Drupal site. It's free for a single site, and only 2 bucks a month for each additional site you want to monitor.
You can use droptor on any shared host or VPS host, but configuring your site to perform all the optimization tasks suggested by Droptor may require higher level access than you can get on a shared host. As always, if you're concerned about acheiving top-notch Drupal performance and configurability be sure to look into VPS and Cloud Drupal Hosts for your site.
Droptor requires a subscription to their service (free for 1 site as mentioned above) and installation of the companion module, at http://drupal.org/project/droptor. Once you've installed the module and configured your account on droptor.com, you can view a slew of security, performance and trending data along with tips on how to maximize your site's security and performance.
Droptor really comes in handy though, when you have numerous Drupal sites to monitor. If you have a dozen or so sites, it's far easier to check a single dashboard on droptor.com to see a snapshot of the health of all your sites, rather than logging in to each site individually to check out a status report.
Once it scans your site, Droptor provides a detailed breakdown of how your site fairs when it comes to Drupal best practices for security and performance, offering step-by-step instructions for how to remedy any problem areas.
Additionally, Droptor offers email alerts for any number of system events, including when modules are available, when certain activity triggers are hit and a weekly report of overall system health.
Since it's free for use on your first site, there's really no risk in giving it a shot. We've been using Droptor for our own sites for a while and are setting all of our client Drupal sites up on Droptor over the next few weeks.
There are several other nice monitoring tools out there, but most aren't as affordable and simple to use as Droptor. For that reason, I'd recommend Droptor to anyone looking to get into site monitoring and Drupal tuning as a cheap, and low-risk entrance point.