Here is the lowdown and the most poplular. . .

JPEG
Is the most popular option and usually the only option on a cellphone of snapshot camera. Generally speaking JPEGs should be used, for personal use, social media, small prints and for sharing on emails, Twitter/Facebook etc.
Benefits are: small file sizes means more can be stored in your selected memory and file transfer times are quicker.
Negatives (excuse the pun) are; suseptable to quality loss due to image compression and not good for image editing, say in Photoshop.

TIFF
This is the most commonly used industry-standard file format, is generally what print or publishers ask for and usually uncompressed. As a result the opportunity for extensive post-processing in image manipulation software is way better.
Benefits apart from extensive manipulation in photo editing software, printing at the highest quality is posible and at much larger sizes.
Negatives are much bigger file sizes which has a slower storage speed and use more space on your storage media.

RAW
Available on advanced compact cameras and DSLRs. This is the best option of all formats and the choice of professional photographers.
Raw files are compressed using a process that retains all of the information originally captured. This means that Photographing in raw format will require plenty of memory cards, not to mention considerable post-processing time. It will also require some basic knowledge of image editing software such as Adobe Lightroom or Photoshop, as files will have to be edited and converted before they can be used (to share online, print, send to friends, etc.).
Benefits are that the best quality image file is captured. Adjustments such as white balance, exposure, contrast, saturation, sharpness can all be altered in an image editing software, after the image has been taken to amazing lengths.
Negatives are the time needed to convert and edit photos. Be prepared to buy lots of storage space as these use a LOT! Post-processing times are longer even on the fastest machine.