What are the differences between the various file types e.g. jpg (aka jpeg) and tiff?

There are only two kinds you need to worry about at the moment so we will forget about the rest for now.

The first is the jpg (JPEG) type.

In order to save space on a disk this kind of file uses a clever calculation to work out which bits it needs to remember and which bits it can forget. It then reconstructs the file based on its calculations. This way of saving a file is know as lossy because it loses some bits of information when it saves the file. These bits are generally not important bits and make no difference to what you can see on screen or print on a page. The vast majority of digital cameras store their images in the format by default.

The second is called tiff.

This file also uses a clever calculation but it doesn't lose any of its information. There is no advantage to this unless you are planning to do some restoration work on the file later (where there may be some hidden information that you can retrieve) or if you intend to greatly increase the size of the image. The big disadvantage to tiff is how big the files are (they are huge!!!). This means you can't save many images to a DVD.

What is the best option?

For 99.9999% of the time jpg is the best option but just occasionally you may want to use tiff.

The other file formats do much the same as one of these with varying success. The only other format is RAW which doesn't compress the file at all but let's not worry about that because you won't need to bother with RAW unless you are a very serious amateur or professional photographer.