if you want to make a wallpaper use PNG(cause its a perfect lossless format that produces a near perfect copy of the workpeice) but its a really huge file because of that.
if you want to make animated wallpaper i would suggest using APNG(if your computer supports animated PNG's. if not use GIF(note gifs are lossy)) they hold only 255 color variations. but i think there are programs online that allow you to make .avi,/.mpg,other video files as wallpapers.
for normal applications just use jpeg and bitmaps(holds more color and uses minimal compression)