This post is exploring different file type, pros and cons of them.
first: GIF
These are mainly used for small graphics, they are the smallest type of file there is because they unlike most use only 256 colors, however they are fast loading, so they are best for small visuals like button, banners, ect. the cons of a GIF are they can be used for photos or anything with a large color range.

PNG
For maintaining original quality with in a small file, PNG is best, with varying degrees of transparency and millions of colors, used best for graphic images, info-graphics and logos work well but not all software or applications work with it, it might complete things if your wanting it for a widespread format file. perfect for a format that uses millions colors but small images.

JPEG
For me JPEG has always been what i used for images, nearly every device and programs work with JPEG, it has a full spectrum of colors, they are perfect for keeping file size small and easy to use, however the quality of the images can be lost when images are edited and saved. JPEG is ideal for displaying photos online but when it comes to you continually editing files its not the format you need.
PSD

TIFF
For bitmap images that you plan to edit continually its what you need, but it doesn't compress into smaller images, TIFF are meant to preserve the quality, they can compatible with photo manipulation programs like photoshop. they offer options to use tags, layers and transparency, therefor for wed format files it isnt for you but for editing digital images that are in a working storage format its perfect.