What tool do you use to monitor your servers?

For a more comprehensive list of monitoring tools and their features, check out this Wikipedia page.

As the question states, what are the most commonly used tools used for this task and what are their strengths and weaknesses?

I've used Nagios in the past with success. It's very extensible (over 200 add-ons), relatively easy to use and lots of reports. A negative would be the initial setup.

Cacti is a very good web-based frontend to RRDTool, providing very handy graphs and stats. RRDTool is the part that gathers data from multiple systems and monitors a wide range of technical data.

We're using that cacti/RRDTool solution to monitor Unix and Windows systems. We get a lot of useful metrics including load, CPU/RAM usage, HD space, users logged in, network traffic, running processes, and so on.

You will find more information on cacti on the What is Cacti? page.

Personally, I love Munin which is very easy to install and to write plugins for as it has a very straightforward architecture. There are quite many plugins already around for all the purposes you could imagine, so you probably won't even have to write plugins in the first place.

It also provides beautiful graphs and the option to configure (very basic) alerts.

Zabbix. It's open-source, and reasonably simple to setup and customise. We have a lot of custom monitoring scripts that feed into the zabbix server, but it takes care of centralising that data, displaying it appropriately, notifications (email, IM, SMS, twitter, etc), and so forth.