TIFF: This is a flexible format that can be either lossy or lossless. There is no external software to change the compression from lossy to lossless. Everything is internal. The file sizes are usually very big. Big digital cameras typically use this type of file.
PNG: This is a lossless storage format which means that it looks for patterns in the image to compress and copy over a larger scale. Once compressed, it can completely convert back to it's original format, if needed. PNG works best with photos with large areas with uniform colors but more than 256 colors in the image.
GIF: This format can support up to 256 colors from a pool of 600 million. If an image has more than 256 colors, it uses the nearest color to represent a pixel. If less than 256 colors, it creates the image exactly. GIF files compress in two different ways. First it can reduce the number of colors. Secondly, it replaces commonly used patterns. GIF can sometimes lose 99.998% of colors in color rich images. GIF files are best used with images that have less than 256 colors. They can also be used to support short animations.
JPEG: This format is optimized for photographs. They have the ability to keep the image smooth and clear while keeping the file size minimal. The degree of compression is adjustable from the standard 24 bit scale. JPEG files are very common across the internet.
RAW: These files are used by some digital cameras. They have smaller file sizes than TIFF which means they have smaller download times. RAW files are used differently by different manufacturers so editing can be harder.
BMP: Uncompressed files that are used by specific publisher. They are unnecessary.
PSD/PSP: These files are used for specific program graphics. For instance Photoshop uses PSD and Paint Pro uses PSP. Editing is very easy. They support multiple layers on a single image.
When working with concept art and art for advertisement I would mainly use PSD, JPEG, and PNG files. I'd used PNG files in the early stages of development and while I'm in the process of still designing the project. I'd use JPEG for online artwork. I'd use PNG for posters and promotional art.