What is DDS?
DirectDraw Surface, which is how game textures are stored for the graphics card. Block-compressed so the GPU can read it without unpacking, and usually carrying a whole chain of mipmaps. Open a game's files and this is most of what you'll find.
What is JPG?
JPEG is where photographs live. Its compression is lossy - it throws away detail the eye is bad at noticing, and throws away a bit more every time you re-save - which buys huge size savings on photos and makes a mess of screenshots and text. There's no transparency in it at all.