This thesis investigates minimal generating sets of ideals defining certain nilpotent varieties in simple complex Lie algebras. A minimal generating set of invariants for the whole nilpotent cone is known due to Kostant. Broer determined a minimal generating set for the subregular nilpotent variety in all simple Lie algebra types. I extend Broer's results to two families of nilpotent varieties, valid in any simple Lie algebra, that include the nilpotent cone, the subregular case, and usually more. In the first part of my thesis I describe a minimal generating set for the ideal of each of these varieties in the coordinate ring of the Lie algebra. My goal in the second part is to describe which images of generators remain necessary when the variety is intersected with a Slodowy slice to a lower orbit and which become redundant, information that can be used to give new proofs of the singularities of minimal degenerations of nilpotent varieties.