JPEG files are typically photographs. They show continuous colour variations and tones throughout the image. Vector files, though, generally emphasise line and shape instead. They are often clip art illustrations, or outlined art like maps. JPEGs can not be enlarged without degradation to the resolution of the image, whereas vector files can be enlarged without any loss of quality. There are programs (like Genuine Fractals) which use complicated algorithms to enlarge JPEGs as if they were vector images, but there are ways to approximate the same result in Photoshop.
Decide what part (shape) in your JPEG you want to convert to a vector file. You will not be able to convert every detail all at once, so you will have to pick out specific objects one at a time. Even dedicated tracing software (e.g., Adobe Streamline) is limited in how much detail from photos can be turned into a vector image all at once. For the sake of this article, let's say you wish to turn a hat from a photo into a vector image.
Increase the contrast in your JPEG. With your JPEG open in Photoshop, select Image, then Adjustments, and the click Brightness/Contrast. Make sure the "preview" option is checked, and then drag the Contrast bar slowly to the right. You are trying to separate the hat tonally from whatever is surrounding it. You may have to play with both the Brightness and the Contrast sliders, depending upon the hat and the background.
Use the Magic Wand tool from the Tool Bar. Try setting the tolerance to 10, and click the hat. If the image was fairly contrasting, it should immediately select the entire hat. If it selected more than the hat, decrease the tolerance and try again. If it selected less than the whole hat, try increasing the tolerance. Use the Magnetic Lasso tool from the Tool Bar, if the Magic Wand does not work.
Make a working path from the selection. Go to the Paths palatte (Window, then Paths), and choose "Make work path from selection."
Export the path. Go to File, then Export, and then click "Paths to Illustrator." Choose a name for the file to save it under, and then click "OK." The file will be saved as an Adobe Illustrator (.ai) vector file, and can now be manipulated in vector illustration programs like Illustrator.
Follow the steps outlined above to convert other elements from your original JPEG file.
Using a dedicated image tracing program is your best and easiest option. If you have Illustrator, the program contains a fairly good tracing tool.
Do not expect to end up with something that looks like your original photo. At best, you will be recreating what looks like a line drawing of the photo.