Is it just the case fan? If so, those are pretty cheap and easy to replace. And they're pretty important to have too. You know, you don't want all that warm air trapped in there. 

Replace it, they're cheap enough.

I think so. Like, my desktop is on my desk and the left hand side is facing me. I can see through a plastic clear pane all the parts, and there's a fan that is attached to this plastic siding. The other two are black and different looking--one says intel. I think its the one on the plastic that's having the issues.

Yeah, I'm gonna say that's a case fan. I think the one that says Intel on it would be your CPU fan. As for the other one, there aren't enough details to go off of. 

If it's a fan that's attached to the case (and that seems to be it since you say it's attached to the plastic siding), it's a case fan. Should only run you about $5 to $15. You could spend more if you have expensive tastes, but you don't need to go huge on a decent case fan. 

Yeah, the other fan is black similar to the intel one but it's much larger and it's directly in the back of the desktop case if you were looking at the back of the desktop. The one I think has the issues is resting on the left hand side where the vents are for air to escape.

If its the one on the side of the case, it may have been knocked or has had a screw come loose. Maybe try just refitting that fan before buying a new one. My fans used to vibrate the top of my case when on 12V but I just added some rubber washers to where the screws are and now it doesn't do it.

Yeah, maybe it got loose or something. Because when it's making that noise the fans aren't going super fast it seems I can almost see a slight delay. Hard to explain. And when I turn off the PC when they finally stop spinning it's a chugging sound almost.