Short answers, pulled from the story.
Noether's theorem states that every continuous symmetry of the action of a physical system with conservative forces has a corresponding conservation law. Published by Emmy Noether in 1918, it connects symmetries such as time-translation and rotation to conserved quantities such as energy and angular momentum.
Emmy Noether published the theorem in 1918. She began the work in 1915 while assisting Felix Klein and David Hilbert with problems related to Albert Einstein's theory of general relativity, and had the key ideas in place by March 1918.
Noether's theorem explains conservation of energy as a consequence of time-translation symmetry, conservation of linear momentum from space-translation symmetry, conservation of angular momentum from rotational symmetry, and the center-of-mass theorem from invariance under Lorentz boosts.
The Noether charge is the conserved quantity associated with a symmetry in Noether's theorem. The Noether current is the flow that carries that charge, and it is defined up to a solenoidal, or divergenceless, vector field.
The quantum analogs of Noether's theorem are the Ward-Takahashi identities. They yield further conservation laws in quantum field theory, including conservation of electric charge from phase-factor invariance of the complex field describing a charged particle.
Noether's theorem does not apply to systems that cannot be modeled with a Lagrangian alone. Systems involving a Rayleigh dissipation function, which models dissipative processes, fall outside its scope; such systems can have continuous symmetries without a corresponding conservation law.