(tl;dr version: if you have access to iPhoto, give it a try–it is an excellent tool)
A few thoughts on this…
Most cheap and cheerful tools will allow you to open file1234.jpg, change it, and then save it again as file1234.jpg.
This means that you lose the original photo, and you will be degrading the quality of the photo each time you open it and re-save it.
This is not normally a problem for folks who just want to clean some stuff up before printing out photos of friends and family, but it is something that you should consider.
If you want to use a basic editor like this, you may wish to make the originals read-only before opening them, and then save the photos with a different name (e.g. photo1234.jpg–>photo1234-edited.jpg).
Beyond simple editing, the next problem that photo tools address is how to deal with all of your photos: how to catalog them, sort them, rate them, and find them and so on. Like iTunes for photos.
This is normally referred to as Digital Asset Management.
And yet another cool kind of photo application is the nondestructive editor. Lightroom and iPhoto/Aperture are the poster children for this feature. The new Photos app in OS X will also have to support this.
If you have access to a free copy of iPhoto, give it a try. It combines a good digital asset manager with a very nice nondestructive editor.
If you use iPhoto, the changes you make, such as straightening, cropping, and some retouching are all tracked as a list of changes, but they are not saved into the source file. When you open iPhoto, you see a rendered version of the photo with the tweaks (which you can print and/or email to people), but you can always access the original. This means that no matter what you do, you can always edit more next year, and you can always go back to the original file. They don’t clog your hard drive with extra copies either: the only thing saved is the list of tweaks (e.g. set WB to 2700K, increase exposure 1 stop, crop to 5x7).
These DAM+Nondestructive Editor tools like iPhoto/Aperture and Lightroom are very good and are often the bread and butter of serious photographers. I spend most of my time in Lightroom, with very little need for true pixel-based editors such as Photoshop. These tools generally handle all of the basic adjustment needs of a pro: there are wedding photographers out there doing serious work from week to week using nothing more than Lightroom or Aperture.
Photoshop and its brethren are far more complex to work with than the simple editors in iPhoto/Aperture/Lightroom, but can do wonderful things. Nevertheless, I would steer clear of these tools if possible in the beginning.