PNG
Portable Network Graphics (PNG) is a bitmapped image format that employs lossless data compression. PNG was created to improve upon and replace the GIF format, as an image-file format not requiring a patent license.
PNG supports palette-based (palettes of 24-bit RGB colors) or greyscale or RGB images. PNG was designed for transferring images on the Internet, not professional graphics, and so does not support other color spaces (such as CMYK).

JPEG
JPEG is at its best on photographs and paintings of realistic scenes with smooth variations of tone and color. In this case it usually performs much better than purely lossless methods while still giving a good looking image.

GIF
Graphics Interchange Format (GIF) is an 8-bit-per-pixel bitmap image format that was introduced in 1987 and has since come into widespread usage on the World Wide Web due to its wide support and portability.
The format uses a palette of up to 256 distinct colors from the 24-bit RGB color space. It also supports animations and allows a separate palette of 256 colors for each frame. The color limitation makes the GIF format unsuitable for reproducing color photographs and other images with continuous color, but it is well-suited for more simple images such as graphics or logos with solid areas of color.

PNG vs JPEG
PNG is a better choice than JPEG for storing images that contain text, line art, or other images with sharp transitions. Where an image contains both sharp transitions and photographic parts a choice must be made between the large but sharp PNG and a small JPEG with artifacts around sharp transitions. JPEG also does not support transparency.

PNG vs GIF
On most images, PNG can achieve greater compression than GIF. PNG also gives a much wider range of transparency options and a much wider range of color depths than GIF, allowing for greater color precision, smoother fades, etc. However, GIF supports animation while PNG does not.

PNG images are widely supported, but not as widely supported as GIF images.

Vector Graphics

What is a Vector Graphic Format?
Vector graphics (also called geometric modeling or object-oriented graphics) is the use of geometrical primitives such as points, lines, curves, and polygons, which are all based upon mathematical equations to represent images in computer graphics. It is used in contrast to the term raster graphics, which is the representation of images as a collection of pixels, and used as the sole graphic type for actual photographic images.
The main advantage is that they are rather small files compared to high-quality raster (pixel) images. They contain graphics that can be almost infinitely scaled in size without degrading the output because they are vector based (resolution independent).

Vector graphic file formats

SVG
Scalable Vector Graphics (SVG) is a text-based graphics language that describes images with vector shapes, text, and embedded raster graphics.
SVG files are compact and provide high-quality graphics on the Web, in print, and on resource-limited handeld devices. They contains graphics that can be almost infinitely scaled in size without degrading the output because they are vector based. In addition, SVG supports scripting and animation, so is ideal for interactive, data-driven, personalized graphics.
SVG is a royalty-free vendor-neutral open standard developed under the W3C (World Wide Web Consortium) Process, moreover SVG is interoperable.

EPS
Encapsulated PostScript, or EPS, is a DSC-conforming PostScript document with additional restrictions intended to make EPS files usable as a graphics file format. In other words, EPS files are more-or-less self-contained, reasonably predictable PostScript documents that describe an image or drawing, that can be placed within another PostScript document. The EPS format contains graphics that can be almost infinitely scaled in size without degrading the output because they are vector based.

SOURCES
You may find some additional information by consulting the sources that were used for this guide. Please find the links to some of these here below:

PNG
http://en.wikipedia.org/wiki/Portable_Network_Graphics
http://www.w3.org/Graphics/PNG/

JPEG
http://en.wikipedia.org/wiki/JPEG
http://www.jpeg.org/

GIF
http://en.wikipedia.org/wiki/Gif_animation

Vector Graphics
http://en.wikipedia.org/wiki/Vector_graphics

SVG
http://en.wikipedia.org/wiki/Scalable_Vector_Graphics
http://www.w3.org/Graphics/SVG/

EPS
http://en.wikipedia.org/wiki/Encapsulated_PostScript