vectro graphics

My girlfriend has just started her own business, and is in the process of having her web site built for her. But we have been asked to supply her logo (which i designed in windows paintshop)in a vectro graphic format. Only being a novice on computers, I dont know where to start. So can anyone help me by telling me what software I might need to do this, how easy it is and any other relevant info about coverting an Image designed on Paint into a vectro graphic format.
Thanks for any help received.