AI and EPS are both vector formats,
PDFs I +THINK+ can contain vector as well as raster images.
TIFs are typically raster. I don't believe they can contain vector images.
for AI and EPS, yes, both these formats +can+ container raster images. Imagine going into a draw program like Illustrator and pasting a screenshot into the image. Yes, it saves as AI, but that image is raster.
As for saving a vector image as JPG, no, I don't believe that's possible. JPG is really intended as a lossy raster format.
Oh, and WMF files (windows metafile) is another example of a vector format (albeit a quite old one).
Finally, Inkscape is a very popular opensource vector drawing package (a bit like illustrator).