Law-abiding means that some equation holds, for example a monoid instance should be associative, so ((a <> b) <> c) should always produce the same result as (a <> (b <> c)). There is nothing checking that my implementation of <> abides by this law, but other programmers (and optimisers or compilers, maybe?) can make use of it, and might write incorrect code if my instance does not abide by the law.

Another example would be functors: it should be the case that (fmap f) . (fmap g) is the same as fmap (f . g), for all functions f and g that make sense in the equation (output type of g must match input type of f).