If you just imported that image into Illustrator and then saved it as .eps, you would not have a true vector. There are a number of stages to go through to make it a vector.
To see how the paths work, try opening a new document, click File, then Place, and put you picture into the document (Better resize it to about 600 pixels first)
Then click on Trace, and then Expand. You can now see the paths that make a picture a vector rather than a jpg.
Autotraced images are not accepted on a lot of sites because they are usually a mess, with loads of confusing paths. You need to use the pen tool to create accurate paths.
Drawing in Illustrator is not like drawing in an art package, it is a totally different approach.