Below I provide more information about different kinds of files, but from a client's perspective I believe the most important thing is what will give them the very best final product.  For me that is capturing the picture in RAW format and doing post processing of the RAW file to either a TIFF or JPEG file format. 

RAW  (Raw Image Format)  This file format is not directly printable and is considered similar to the film negatives.  As with film negatives, a RAW digital needs to be processed to the equivalent positive print image.  Camera makers each have their own proprietary structures for these files and a special piece of software (RAW converter) is required to process the images.  

The primary value in capturing pictures in RAW format is the huge flexibility after the picture was taken to make adjustments and corrections to the picture, just as professional film photographers use to 'tweak' their prints during the development and printing phase.  Instead of doing this in the darkroom with solutions and chemicals, this work is now done extensively on the computer.

TIFF (Tagged Image File Format)  This file format has been around since the mid 1980's and allows for high quality, color depth images.  While is has lots of flexibility, one of its most useful capabilities is that changes can be made to the image and re-saved without losing image quality.  It is the equivalent of a print like image rather than a negative.

JPEG (Joint Photographic Experts Group)  This is one of the most common file formats and allows for compression of an image from a large file to a much smaller file.  This makes them easier to post on the internet and email to friend as they tend to be much smaller in size than either a RAW, TIFF or DNG file.  While high quality JPEG files are commonly used for printing, extensive editing and re-saving the file will result in loss of image quality.  In addition, the types of editing possible with JPEG file format is less than either RAW, TIFF, or DNG formats.  

DNG (Digital Negative)  This is a format promoted by Adobe and is considered to be a 'generic' form of a RAW image.   

PSD (Photoshop Document)  This is another Adobe image file format tailored to be used with it's digital image software suite.

There are many other file formats, but the five mentioned above are some of the more popular ones for photographers.  Other formats include GIF (Graphics_Interchange_Format)  and PNG (Portable Network Graphics)