I understand the concept of symmetry in geometry, but what do theoretical physicists mean when they speak of symmetry?
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A symmetry describes any kind of transformation that maps back to the original state. If you have a mirror symmetry, then performing that mirroring doesn’t change the image. If you have a rotational symmetry then rotating by the corresponding angle doesn’t change the image.
In physics this works similar: Spatial translation symmetry means that no matter where you are, the rules of the universe are the same. Spatial rotation symmetry means that no matter which direction you look, the rules of the universe are the same. Time translation symmetry means that no matter how early or late you look, the rules of the universe are the same.
There are also some symmetries that aren’t true for our universe. Parity symmetry means that if you mirror space in one dimension, the rules of the universe stay the same – but the weak interaction isn’t parity symmetric. Charge symmetry means that if you replace every particle with its antiparticle, the rules of the universe stay the same – but that’s not the case either.
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