CPU: As the processor temperature keeps increasing, for every degree, its life keeps getting shortened by the heat. Therefore, making sure that your CPU is running within a safe temperature range is of utmost importance. Again, most modern distros will bundle in some form of driver which will interact with the temperature sensors placed inside the hardware. There are various methods of tapping into the information provided by the sensors.
If you have a modern ACPI-compatible system, and chances are that you do, you will be able to monitor your CPU temperature easily. Typing "acpi -t" in the command line will give you the processor temperature. If you get an error saying that the device is not supported, you most probably have an incompatible motherboard.
In such cases, lm_sensors comes to the rescue. lm_sensors is the de facto hardware monitoring tool for Linux. It's capable of monitoring most, if not all the sensors present in your computer to give you a complete picture of how your machine is working. Again, your specific distro's package manager will come to the rescue.
Hard-Disk: The hddtemp service will give you your hard-disks' temperatures through the SMART functionality found on all drives today. All you have to do is install the service if it's missing and run it. It will run in the background and monitor your disk temperatures. To check the temperature, a simple "sudo hddtemp /dev/xDy" (where x can be h or s, and y is an alphabet. for example, /dev/sda or /dev/hda) will suffice.
Motherboard, graphics card, etc: lm_sensors to the rescue here. Just make sure you have I2C support in the kernel; Hardware Monitoring support and relevant hardware monitoring drivers compiled as modules. Once installed, a simple "sudo sensors" will give you the information you need.
Frontends: Chances are, you are using a graphical interface everyday and you would rather have the information presented to you in a shiny application rather than the boring command-line. And luckily, graphical frontends for these sensor applications are available by the hundreds; from a simple Gnome panel applet to full-blown remote-monitoring applications. Personally, I use the computertemp Gnome applet which sits in a panel and taps into all available sensors. For a much bigger list, make your way here and scroll down to the Hardware monitoring applications using the lm_sensors package section