There are many different file formats that you can save your photos in. I'm going to talk about a few of them below.

PSD files is Photoshop's native file format. It supports everything Photoshop does.

JPG files is the most popular format. It is the most universally accepted format. It is compressed so when you save as a JPG don't open and resave the file. The file will get drastically worse each time you do that. Use this format once you are done editing the file. It doesn't support things like layers and channels.

TIF files is a standard image format that is half way between JPG and PSD. There is many different kinds of compression in TIF files. They can either be lossless or lossy compression.

There are a couple different other file formats that are relatively new and hopefully one of them will replace JPG files. They are Microsoft's PhotoHD format and JPG2000. I have played around with the PhotoHD format and I think it really has potential to replace JPG as the standard image format.