A research grant is funds to perform research given to professors by the government or private institutions. Professors will write a research proposal for the funding agency in order to apply for the money. These proposals are reviewed by scientists and the top proposals are funded. The grants cover cost of salaries (for the professor, graduate students, postdoctoral fellows, and research staff), laboratory equipment and supplies, and overhead (i.e. the university takes a percentage of the grant to support the department).
As for working in a research lab, you'll likely be assigned to work with a graduate student and postdoctoral fellow and you'll be helping him/her with their research project. They will train you in the techniques you need to know and give you instructions on what to do.