This is for a specific type of common fan, but the steps can somewhat be applied to other similar fans.
The fan I had sounded like an angry alien spaceship taking off. Usually this much noise means bad bearings, and sometimes from foreign debris, like dust buildup or small animals or insects nesting in the housing. Mine was the common worn out bearings.
First, safety. Normally I'd say "you are on your own to find out how to be safe, and please don't do anything without full knowledge of what you are doing, be careful of electricity, sharp edges, falling, social media trolls, stray meteorites, etc." I mean all that today, but perhaps more importantly, this: DO NOT DO THIS WITHOUT COVERING UP YOUR WORK AREA IF SAID WORK AREA IS THE KITCHEN TABLE. I can't stress this enough, your significant other just may cause you some kind of harm if you make a mess in the place where you also eat stuff. If you don't heed this warning, man, you are on your own.
Second, find out what kind of motor you have. You'll need to take off the cover and look at it. Some covers are held on with springs, others like mine with a small nut. Once you can see the fan, do not touch it until you can be certain there is no power going through it. This one has a simple 2 prong plug to disable it. Make sure the switch is OFF before you pull this plug (yours may be different, if so, good luck)
Third, use a search engine to find out what the fan might cost. I usually add about $10 US to the web cost of a thing for shelving upcharges at your local store, or if shipping is free and I can wait, just buying it.
Step 1: 1. Take That Old Fire Hazard Down!
Remove the old fan. This induction motor is taken out by unplugging the power and loosening the two screws on the bracket. Simply wiggle it a little to work the whole thing out of the housing, fan blades too.
NOTE: SAFETY. Be careful that you have good footing, use a proper mini ladder or tall stool designed to stand on. The toilet seat lid can break, it was not meant to be stepped on ever.
That said, I used the toilet lid just fine and didn't harm it. Use at own risk. Just sayin'.
Step 2: 2. Get the Bracket Off.
Take pics of the fan motor with your smart phone. You will want to remember where the plug is in relation to the bolt that holds on the grill. In a later OOPS! step, I'll show you a slightly time consuming fail. Don't do the thing I did there.
Undo the two little nuts that hold on the bracket and keep them safe, you will need them later.
Pull of the bracket. It will probably come off easily.
Clean the greasy guck off the bracket.
You are doing this over a disposable surface, right? Wow, if not, you are brave.
Step 3: 3. Unbox Your Replacement Fan
You can buy just the motors for cheap, but I bought the whole installation for about the same price at a local home improvement store. The motor was installed in a metal housing similar to the one already in my ceiling.
Can you just install the whole brand new housing? Yes. Is it easier? Not really.
The old housing should be similar in size and shape. If it is close like mine then you can just install the motor back into the old box already in the ceiling. If you swap out the whole box, you might need to cut into the drywall on the ceiling, do some wiring, paint, etc. Extra steps, more work, less fun.
Make sure the fans are the same specifications. The two little bolts that hold on the bracket on to the motor should match up with the two little bolts on the new fan. Undo these and remove the fan from its housing.
You may need to pull the fan off first. Just pull. The fan should slide off the motor shaft with a little effort.
Step 4: 4. Attach the Bracket to the New Motor (with OOPS!)
I took the small long bolts out of the motor and ran them through the bracket and back into the motor. I forgot to reinstall the grill mounting bolt. I loosened up the bracket and fit the bolt through the hole, then re-tightened the bolts.
Also, I forgot to orient the bracket the right way (there are two ways it can go on, the way I first did it, and the correct way). So, when I went to reinstall the motor, the plug wouldn't reach. Make sure the motor is on the right side of the bracket.
Step 5: 5. It Should Look Like This..
Last step is to press the fan back on the motor, put the motor in the housing and tighten those two larger screws on the bracket.
Plug it in. Flip the switch and smile. The little beastie should work without making a noise like an injured cat!
NOTE: do NOT injure a cat to compare the sounds. They don't like it.
Rinse off and dry the grill cover and screw it back on.
Extra step (mostly for guys): Stand there and flip the switch a few times to admire your handiwork while grinning like a fool. Go ahead, you've earned it! Ok, now stop that.
Step 6: Disposal.
The leftover parts from the motor are cool looking, but don't keep them, they are useless without a motor. Put them up on 'free stuff' on the CL, break them down for recycling, whatever. DON'T keep them. They are really useless. Seriously.
Ok, the grill is pretty sweet, maybe you can...
NO! Just say no. Get rid of it, all of it. It is rubbish.
Any nice comments will be welcome, including suggestions. Trolls are ignored.
Hope this helps you!