For all of you designers out there that don't understand exactly what is the difference in "lay" terms of the most common web image formats I will go on with a little explanation and examples for you:
The background is a PNG image that is always the same, and the images are set over it

PNG

This is a relatively new file format, it stands for Portable Network Graphics, and it's primary characteristic is that it has lossless data compression, meaning that it won't lose any quality whatsoever.
And the other greatest thing about it is that it has full alpha support, meaning that it can have different opacities and transparencies.
Here are below some examples of PNG images with different backgrounds:

Dice in PNG format
Dice in PNG format
Dice in PNG format
Dice in PNG format

The size of the PNG file on top (100x50px in a row of 4) is of 12,4Kb
As you can see it can be perfectly seen even though the background is changed (stripes, solid color, alpha transparencies bg, and transparent bg)

GIF

GIF image format is probably the most common one in the web, or at least it has been for the past years. It stands for Graphics Interchange Format, and it has 3 characteristics that make it unique and sometimes useful:

  • It has lossless data compression (Pretty much like PNG)
  • It supports animations, although they increase a lot file size
  • It supports transparencies, although it doesn't support alpha opacities.

Here is the same demonstration as before, but now with GIF graphics on top:

Dice in GIF format
Dice in GIF format
Dice in GIF format
Dice in GIF format

In order to look good GIF images with transparency often use a surrounding border, that should match the background color. As you see in the example the image looks just great when displayed over white, but when displayed over other types of background colors the transparency looks awful.

A great advantage though is that this GIF image weights 4,04Kb, against the 12Kb of the PNG file...

JPEG

JPEG stands for "Joint Photographic Experts Group" and it is not the actual name of the format, it is the name of the comitee that created it, but since everyone decided to call it this way, lets stick to it ;)

JPEG offers some characteristics that are worth looking at when designing for web, it uses lossy compression, which basically means that when saved it will lose quality.
Depending on its use this may not be bad. For example in this site's design I've used JPEG files for backgrounds, since the quality loss contributes to the "rustic" effect, but it should never be used for logos or sharp things.

The good use for JPEG is for large photographs, big backgrounds, or other large images.
Here is an example of the dice image, using different compression quality:
Since there is no transparency the background will be seen always as white...

High quality:

Dice in JPEG format (High quality)
Dice in JPEG format (High quality)
Dice in JPEG format (High quality)
Dice in JPEG format (High quality)

Medium quality

Dice in JPEG format (Medium quality)
Dice in JPEG format (Medium quality)
Dice in JPEG format (Medium quality)
Dice in JPEG format (Medium quality)

Low quality

Dice in JPEG format (Low quality)
Dice in JPEG format (Low quality)
Dice in JPEG format (Low quality)
Dice in JPEG format (Low quality)

File sizes were:

  • High quality: 31,2Kb
  • Medium quality: 26,2Kb
  • Low quality: 25,0Kb

Conclusion

When designing you should try to use GIFs as many times as possible, since they provide the smallest file sizes, and better quality than most JPEGs. In the cases where you need alpha transparency you should consider changing to PNGs, but avoid their use, they'll increase a lot page load time...

Have fun designing ;)