Noether was indeed a woman, first name "Emmy".
Examples of the theorem: If the Lagrangian is invariant under rotations,
one gets the conservation of angular momentum, invariance under
translations gives conservation of momentum. These days one expects
Lagrangians invariant under various quantities which would yield
conservation of charge, isospin, hypercharge and the wierd things one
finds conserved in the reactions of subnuclear particles.