What business license I need to run a mobile car wash (cleaning business etc.)
Florida does not have a state-level license for this type of business, but you are likely to need a business tax permit (formerly occupational license) from your county tax collector. A list of Florida tax collectors with contact information is available here: http://floridarevenue.com/dor/property/taxcollectors.html. If you are within a municipality, you may also need an occupational/business license from it.
The State Library offers a guide to starting a business in Florida (http://dos.myflorida.com/library-archives/business/). This page has links to required taxes, how to incorporate, and more.