A symmetry under X means the system under test is unchanged (ie the same physical laws work, your predictions are still correct) when you do X.
A simple example is the symmetry under spatial translation -- if your experiment still behaves the same way if it's moved a meter to the left, it has "spatial translational symmetry". This symmetry isn't exactly true on the surface of the earth because of variations in the gravitational field etc., but on a small scale for lab experiments it's true, and in deep space it's certainly true. Another example is symmetry under spatial rotation -- your experiment doesn't care whether you face it north or east.
By a very cool bit of maths called Noether's Theorem, you can show that for every symmetry that a system has, there is an associated conserved quantity. So systems with spatial translation symmetry will show conservation of momentum. Systems with time translation symmetry exhibit conservation of energy -- within that system, you can't create or destroy energy. Rotational symmetry results in conservation of angular momentum.
Much of modern physics is built around identifying the symmetries that the universe (or parts of the universe) obeys, the associated conserved quantities, and what happens when those symmetries are broken -- for example the maths leading to the Higgs boson. Currently we believe the universe overall obeys C(harge) P(arity) T(ime) symmetry, that is if you change matter for antimatter, flip everything spatially (as in a mirror), and reverse the direction of time, everything would be the same. This recent experiment shows that time symmetry by itself is not obeyed -- if you only reverse the direction of time, this particular particle collision is not the same.