Most people pick an image format by habit. There is a simple rule that gets the decision right in almost every case, and it comes down to what is actually in the picture.
The rule
Photographs go to JPG. Anything with text, sharp edges, flat colour or transparency goes to PNG.
Why
JPG throws away detail to save space, and it is very good at throwing away the kind of detail your eye does not miss in a photograph. Gradual changes in tone — skin, sky, foliage — compress beautifully.
The same compression is terrible at hard edges. Put black text on white and JPG scatters faint grey speckles around every letter. Screenshots and diagrams are the worst case, which is exactly where PNG belongs: it is lossless, so an edge stays an edge.
The transparency question
Only PNG carries transparency of the two. Save a cut-out as JPG and the transparent area comes back as solid white, which is the most common reason a logo turns up with a white box around it.
- Photo, no transparency, size matters → JPG
- Screenshot, diagram, logo or text → PNG
- Anything with a transparent background → PNG
- Going to print at large size → PNG, or the original file