The Pros and Cons of Files

PSD
Advantage: It contains PhotoShop data and can open it where other files can't.
Disadvantage: The file will only open if the computer has PhotoShop installed.

TIFF
Advantage: It is good for printing.
Disadvantage: Some programs will not load TIFF.

JPEG
Advantage: All web browsers support JPEG. It is also small and doesn't take up much space. If you want to print it straight away without using PhotoShop, you can do so easily without converting to another file.
Disadvantage: The picture loses detail when converting it to another file, known as "lossy".

PNG
Advantage: It is good for web graphics
Disadvantage: Not all web browsers support PNG

RAW
Advantage: It is much larger than JPEG and can hold more detail. It can also recover "lost information". For example, if the picture is over exposed, you can easily bring the detail back using PhotoShop. You can't do that with JPEG.
Disadvantage: There are different types of RAW files for each camera manufacturer, so not every camera will take it. Due to its larger size, RAW won't take as many pictures. Not all photo editors will take RAW. You will need to convert RAW file to something else (e.g. JPEG) in order to print.