Long story short:
.bmp: excellent quality, takes a lot of diskspace
.gif: the worst regular quality (of all 4 types), although the lightest one
.jpg: high quality, takes little diskspace
.png: high quality, takes less space than BMPs (and sometimes less diskspace than jpg's)
Formats supporting transparent backgrounds:
PNGs are chosen as HQ images with transparent backgrounds or any transparent areas. (JPGs and BMPs don't support transparency.)
Files in the .bmp format tend to take a huge amount of diskspace, but they are the best ones quality-wise.
Some of BMP types:
R5 G6 B5
A1 R5 G5 B5
X1 R5 G5 B5 <- worse quality, takes less diskspace
R8 G8 B8 <- the most common (normally used) one
A8 R8 G8 B8
X8 R8 G8 B8
GIFs can be easily used as animated pictures, plus they support transparency. The problem is that almost all .gif's are worse than any of above filetypes.
HQ and cheap. Warning, there is a HUGE quality loss when saving with MS Paint. You need to use Photoshop, GIMP, Paint.NET or any program supporting JPG quality levels to save them w/o losing any quality of pictures.
Supports transparency, there is no quality loss when saving with e.g. Paint. Generally HQ pics. Takes a reasonable amount of diskspace, although a deal greater than jpg's.
PNGs can have different compression levels.
16 bits R5 G6 B5: 2,50MB
16 bits X1 R5 G5 B5: 2,50MB
24 bits R8 G8 B8: 3,75MB (the regular one)
32 bits X8 R8 G8 B8: 5,00MB
Take a careful look at enlarged pictures. You'll see the difference between 100/90% JPG and 80/70/60% JPG.
You can download these pics here:
So the best one is 90% JPG, it doesn't show any loss of quality, and cuts the file weight of 100% JPG by about 65%.
You need to use better graphics editors than MS Paint to obtain the best results of file conversion (Photoshop, GIMP, Paint.NET, and so forth).