How do I change the engine radiator fan motors on my 2001 Buick Century?

http://www.rockauto.com/ and search Buick>2002>Century>3.1>Motor Mounts

There are 2 types of engine mounts. The first one rockauto has listed is the easier to change; they are staring you in the face when you open the hood. To change, you'll need a 15mm socket, a ratchet (extensions are good too) and a 15mm wrech. 4 bolts hold the bracket to above the radiator and then 1 bolt goes through the rubber portion. Remove and replace.

The other mount is a little bit more tricky as it is under the oil pan and above the subframe behind the passenger front tire. This can be replaced by dropping the subframe and supporting the engine or hoisting the engine at a slight angle. IIRC those are 15mm as well but might be 17mm.

the transmission mount is on the top of the transmission on the drivers side next to the battery,and one motor mount is under the motor on the passenger side next to the harmonic balancer. and the is either one or two mounts on top of the radiator to the front vavle cover

they are not relays its a blower motor resistor a one piece unit usually held in by two bolts and an electrical connector either under the passenger side dash or behind glove box

Before you did any repairs to address your overheating problem, did you check to see if your fans were working at all ? One way to check if they are working is by starting your vehicle and turning the A/C on, they should automatically start running. If they do not, begin to perform diagnosis to that electric circuit.

test and replace blower motor resistor

On most vehicles, there is a thermostatic switch located on the bottom tank of the radiator that controls the electric fan. When AC is turned on, this switch is bypassed so its failure is 'cured.' These switches aren't terribly expensive nor difficult to replace but ensure you don't lose coolant on the ground while replacing the switch since the stuff is tasty and toxic to animals.
If you like to be able to research and or repair minor problems like this, you might consider buying the Haynes manual for your car. This will also let you approximately access what a repair might cost for labor if you can see the degree of difficulty yourself.
Haynes Buick manual
These are available at auto parts stores often cheaper than the online price. We have on for (and IN) each vehicle we own since they can be lifesavers now and then.

It should be on the right lower side of the radiator when looking at the engine .

Fan will be under dash on passengers side , Removing glove box might be needed, Gm also uses a blower resister ro control fan speeds. First check your fuse and then check and make sure your getting 12 volts to the terminal at the blower, You could have a blown fuse, a bad switch, or a bad resister , if your getting voltage to blower motor then most likely it will be the motor. These are not to difficult to change. I hope this helps, please be aware I can only guess at what the problem is from my past experience.

Sounds like a problem with the "Drivers side temp actuator motor".
Each side has a motor that controlls the temp. door inside the dash for the driver or passenger side. These motors are approx. $75, plus 1 hr. labor to install. See more info here...
http://www.greatautohelp.com/more-buick-questions.html

I have a 1997 Buick Century with the same problem. Both my fans work if the A/C is
on as they should. If your turn off the A/C the passanger side fan will not come on
and at an idle the coolant in the radiator will boil, you can hear it in the overflow tank.
If you are driving there is enough air flow thru the radiator that the coolant doesn't
boil and the car doesn't overheat. I have replace the water pump, the thermostat.
and the ETS or engine temperature sensor mounted just above the thermostat in
the thernostat housing, to no avail, I still have the same problem. The only other
thing is the coolant level sensor in the radiator, which I haven't replace yet due
to the fact you have to pull the radiator in order to replace it, which is a major
undertaking. I have a low coolant indicator on all the time on the dash so I have
a pretty good idea the coolanat level sensor is bad. The coolant level is not low
so that sensor is not working as it should. I hope this may help with your problem.
To test you fan raise the hood when it is cold and top off the radiator and fill the
overflow tank to the cold mark. Start the car and let it idle with the A/C off and
see if the drivers side fans comes on. It will have to get to operating temp
before the fan will come on. If it doesn't come on replace the radiator low coolant
sensor on the passanger side of the radiator. If that doesn't fix the problem
replace the thermostat and the Engine Temperature Sensor at the thermostat
housing with an A/C Delco Engine Temperature Sensor. Good Luck..