You should always have a vector logo file. Why?
If you don’t have a high resolution version of your logo it will look pixelated (seeing the square image pixels making the image look rough), jagged or blurry, especially when blowing it up. If you don’t have your logo in a vector program like Illustrator your logo won’t enlarge to be put on a banner, car or large sign. If you try, it won’t look professional at all. If you don’t have the right file format it may not even look good on your business card.
I prefer to do all my graphic design in vector files, meaning you will always get crisp, clean lines. Your printed and website materials will look their professional best.
If your designer sends you a file with a white background and that’s all you have to work with, what happens when you want to put it on a blue or black background? You’ll have a white box around it. You’ll have to fix it or go back to a designer to have it fixed. You’ll end up spending more time or money if it’s not done right the first time and you don’t get all your files.
Besides vector files (ai and eps) you can also have a PNG with a transparent background. That will work well on web applications.