Fellowships are usually for individuals and are designed to cover 100% of your time - both research and teaching - with the money you receive mainly being used to replace your teaching duties. This is designed to allow you enough time to complete a significant research project and will often be for a shorter duration than research grants. If your research doesn't require a big team or expensive equipment, but does require that you have some time to dedicate to pursuing it, this might be the form of funding for you.
Further details on Fellowships for Early Career Researchers.
Further details on Fellowships for Established Researchers.