There are a few paid apps in the app store for this, but the one I use is
PhotoSync. A free alternative is the Dropbox app. You can set Dropbox to auto upload your photos to a CameraUploads folder and when you sync your Mac with Dropbox the photos will download. Being as it's Dropbox if you have photos you want to send to your iPhone you can drop them in a folder in Dropbox and open the Dropbox app on your iPhone, view the photo, tap on it, then press the button that allows you to save it to your Camera roll. Note that doing that means it's a new photo, so Dropbox will sync the picture back to the CameraUploads folder if you have the automatic photo upload set.
As to folders. The Camera Roll is where everything is saved. Think of it as a DCIM folder for a camera (which actually it is). You can create new albums as a way to view your images in the Camera Roll instead of all images at once. Unlike how it SHOULD function (my opinion) it does not actually move your photos into an album but merely provides a direct link to those images in the camera roll.
Now, if you ever want to jailbreak,
PhotoAlbums+ (which I have) does this properly. It will move photos to albums and delete them out of the camera roll. It also prevents automatic upload of photos to your Photostream if you are using that and set it that way. You can jailbreak if you are still on iOS 6.1.2 or less.