Muse, the good news is that you can crop JPEG images without losing image quality. But first, you have to understand the technology behind JPEG encoding.
I'm not going into full detail here,
since you can easily read about it yourself. But understand this: JPEG is a lossly format. This means that the data stored in the file doesn't contain the exact information about each pixel in the picture, but an estimation and approximation of what human eyes can see. It's just like a MP3 to music. The benefit is that you get a lot smaller file size; the downside is any modifications you make and have to re-encode will cause the quality to decrease. Key word here is re-encode; if you don't re-encode, you will not lose any quality if you save again.
Some very smart people figured out that a
few modifications to a picture can be done without needing to re-encode the image. For example, rotating the image. Instead of rotating the entire image and then re-encode as JPEG, you can rotate and move each block in the JPEG file, and just save without re-encoding the image. Cropping is the same idea. Not a flexible as regular cropping, but a lossless crop can be done.
If you google "lossless jpeg crop", you will find plenty of editors for you. Personally, I use two free editors.
jpeg lossless rotator for a simple and mostly automatic way of rotating jpegs
xnview for anything else. Note that xnview is a lot more powerful tool than just lossless cropping.
FYI, editing jpegs using tools provided by Windows are all lossly transformations. That includes rotating pictures in the picture viewer (whatever they call it).