The positive (equally non-negative) integers form a cancellative semigroup under addition.
Any free semigroup or monoid obeys the cancellative law, and in general, any semigroup or monoid that embeds into a group (as the above examples clearly do) will obey the cancellative law.
In a different vein, (a subsemigroup of) the multiplicative semigroup of elements of a ring that are not zero divisors (which is just the set of all nonzero elements if the ring in question is a domain, like the integers) has the cancellation property.
The cross product of two vectors does not obey the cancellation law.
If a × b = a × c, then it does not follow that b = c even if a ≠ 0 (take c = b + a for example) Matrix multiplication also does not necessarily obey the cancellation law.