Which figure type to use: pdf or eps?

Use PDF. EPS cannot be imported directly by pdftex but must be converted using something like epstopdf. These conversion procedures will often cause unwanted changes to the graphics, such as lossy JPEG encoding of embedded bitmap images. Pdftex will include PDF files directly without making any changes (except for unifying fonts, and even that can be disabled if needed), so you can have complete control over the final result by generating a PDF which is exactly as you want it (assuming your image editing software gives you control over image encoding lossiness, colour spaces, etc).

EPS doesn't support transparency and embeds bitmap images without compression. PDF all the way.

Clearly PDF:

• pdf is an ISO standard, eps is not
• pdf can directly be used via \includegraphics and compiled with pdflatex (see answer by Lev Bishop)
• pdf has more features than eps, i.e. transparency (see answer by Will Robertson)
• even your grandma has a pdf reader on her computer, so if you send her just the images she will be able to look at them
• more software can export to pdf than eps, e.g. Excel can export diagrams as pdf