My stock CPU cooler fan is very loud, can I just unscrew it and replace it with a Pure Wings 2, or do I need a specific CPU fan, or do I need to replace the cooler and heatsink all together? Thanks!
Replace fan, or entire CPU cooler?
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It's likely. Photos taken to a local computer store will confirm for sure and they will be able to sell you something.
The clips require pinching to allow the fan to lift up (if I'm thinking of the right thing). This can be difficult, and I'm not sure if there's a tool available for it (any time I've done it has been disassembly and destruction hasn't been a concern).
The most important numbers are the voltage (probably either 5V or 12V) and the diameter of the fan (so that it fits on the existing mounts)
Edit: I've looked at the specific CPU fan you quote, and it's definitely a non-standard shape. You may have more difficulty replacing just the fan, and may require a new heatsink/fan combo. Note this will be a more intrusive change and will require thermal paste as well.
TLDR available at the end.
You can't replace the fan without replacing the entire plastic housing on most of the stock Intel coolers. The fan assembly is built right into that plastic frame, which you can see by looking into the gap where the wires connect the fan, as well as between the fan itself and that plastic frame where you can see the bearing assembly and the motor.
Here's a picture of the fan after disassembly:
You pull that apart and there's a decent chance you're not getting it put back together. In either case, you'll need to remove the cooler from the CPU / motherboard entirely to make sure that you don't damage any components trying to rip the cooler apart. You don't save any time or work on disassembly by only going for the fan.
There are other problems with this approach as well. Just from disassembly we've exposed either the motor or the bearing assembly which means that dirt and dust can get in there now. You can see that you'll also be required to have some soldering tools and skills at your disposal as well.
You can find cheap used ones on sites like craigslist and ebay for about $10 USD very easily, just get one that matches the motherboard socket / CPU generation you have in your PC.
TLDR: This project is not worth saving $10. Buy a replacement cooler.
How to find a replacement cooler:
If you don't want to buy a used one then you can find aftermarket low profile coolers for $15-35 USD. You don't need anything high end for normal use. Relevant metrics will be:
"cooler height" = how tall of a cooler your case can fit, basically distance from mobo socket to the opposite side of the case
"TDP" = Thermal Design Power (this basically describes thermal dissipation). Your CPU model will have this listed on the specification sheet, so will the coolers. A CPU with a TDP of 95W (watts) will require a cooler with a TDP of equal or greater value than 95W.
"decibel level" = amount of noise generated, lower number is less noise.
"CFM" = Cubic feet per minute of airflow. This is a useful objective measurement you can use to compare fan performance against other fans. Higher CFM = more air being moved.
If a cooler doesn't list these specifications anywhere (check multiple sellers and the manufacturer site) then it is not worth purchasing.
You will also need some thermal paste to install the new (or used) cooler. I recommend Arctic Silver 5 for reliable performance and longevity at a good price. There are better pastes out there in terms of thermal conductivity but they tend to dry out faster on average than Arctic Silver.