Most computer users do not like to troubleshoot fan failures when the messages appear but they should know that the cause of these issues could go undetected for weeks if not months if it were not for the fan failures and they would risk having more serious issues.
Alert! CPU 0 Fan Failure
This warning indicates the Central Processing Unit (CPU) is in danger of running hotter than normal as the fan that is used to keep it cool is failing. Since most computers can be placed in a tower or desktop stance, the stance is most likely in a horizontal position and the vent in the lid of the case is blocked by the monitor and printed documents which is restricting airflow. Once I see such a setup, I will put the computer in a vertical position but after I leave, the client will often put it back to the way it was as the monitor would end up being lower. What they should do is get a stand to raise the monitor and also have space to tuck paperwork.
Alert! Rear Fan Failure
Most people see this message and then replace their Power Supply Unit (PSU) only to see that the error is still present. On a custom-built PC, your case will have a fan in the rear to pull heat out of the case (the correct way) or pull cool air in (the incorrect way). Pulling cool air into the case also pulls in dust, smoke, etc. so always have the rear fan expunge the heat rather than pull cool air in. If your PC was built by a PC company such as Dell, it won’t be the PSU but the fan (often called motherboard fan) that pulls air in from the front of the case to cool the interior starting with the boot drive and a portion of the motherboard.
Alert! Previous Fan Failure
This indicates one or more of the connected fans had issues while the computer was on. To find out which and the time, go into the BIOS and check the reports. When you see which, open up the case and do an inspection and run a diagnostic so that the fans can operate at the maximum revolution per minute and dislodge any debris that may be slowing it down.