Although images can be saved and used in many types of different formats, below are 4 file types, widely used in design applications, and their ideal usage:

EPS format: is a universal cross-platform format.
This file supports images in a vector format and can be modified in color and size without any loss of resolution. This file format works well for press or print situations which may need to resize the image from very small to very large without any loss whatsoever. One of my favorite options is it is able to save your image with a transparent background.

JPG format: is a compressed format used to reduce the size of the file.
JPG files can not be modified in size and color without loosing resolution. This format works well for applications like Microsoft Word, Excel, Power Point and to send via email. This format does not work well for enlargements as it will loose considerable resolution that will result in pixelization.

Tiff format:  is a flexible bitmap image supported by virtually all paint, image-editing, and page-layout applications. TIFF documents have a maximum file size of 4 GB. And can save images in layers with or without a transparent background. It is also a great image file to use when adding a monotone in applications such as Quark Express.

Gif format: this format is supported well by internet applications and is a low resolution image best when used at original size or smaller (never larger). The original colors are converted to RGB. It is preferable to use it in elements designed to be viewed over the internet. This format works well for website and email applications.