The definition of monotonicity is:
$\forall i, x_i’ > x_i \implies (x_1’,\dots,x_n’) \succ
What this means is that if you take a consumption vector (bundle) and increase the consumption of all the goods, the new bundle has to be better.
If the “goods” were bads, increasing consumption of them would lead to a worse bundle. This means that monotonicity is
indeed violated if the “goods” are bads.
Being monotone means that the goods are always indeed, goods, not bads.
Bonus: The first definition I gave is also referred to as “weak monotonicity”.
Strong monotonicity as its name implies, is a property that implies weak monotonicity but not backwards. Its definition is:
$\forall i, x_i’ \geq x_i$ and $\exists j : x_j’ > x_j \implies (x_1’,\dots,x_n’) \succ
This means that increasing consumption of just one of the goods is enough to achieve a strictly better bundle. Many of the commonly used preferences concerning goods that are actually “goods” satisfy this property.
The most familiar exception being the Leontief or Perfect Complements preferences/utility function in which increasing consumption of both goods achieves a strictly better bundle (monotone) but increasing consumption of only one good at the corners/kinks of the indifference curves will leave you at an indifferent bundle (same utility level), hence not strictly monotone.