Mirror symmetry - this is what people typically think of when they talk of symmetry. In this case the base unit is mirrored along an axis producing a copy with its geometry swapped across the axis.
Rotation symmetry - here the unit is repeated by rotating it around an axis. Many plants and some animals like starfish exhibit rotational symmetry.
Translation symmetry - the unit is shifted along an axis one or more times. A chain shows translation symmetry.
Scaled symmetry - in this case the unit is scaled while repeated. A matryoshka doll would be an example.
Symmetries can be combined in various combinations like rotation+scaled (Nautilus)
One theory of why you see so much symmetry in biology is that it takes less information to build something using symmetry than specifying the whole directly. the genes of a centipede do something like specify "here is a body unit with legs" build more until a different gene says "stop". Even then, the gene specifing the body unit is more likely to specify "here is half a segment" and "stick a segment here and repeat that 4 times making each one smaller" to add the leg.
This repetition of units takes less information and results in lots of symmetry as the units are transformed across different axes of symmetry.
Fractals are another example of the repeating action producing repeating units with symmetry.