A state quantity
Entropy is a state quantity: its change between specified states does not depend on the actual route taken. Reversible heat transfer divided by absolute temperature provides a way to calculate changes, with varying temperature requiring integration rather than one universal heat-over-temperature calculation.
References: [1]
Irreversibility and total entropy
When heat flows spontaneously from a hotter body to a colder one, one body loses entropy while the other gains more. The total change is positive. A proper second-law account includes relevant surroundings; observing a local entropy decrease alone does not establish a violation.
Microscopic arrangements
A statistical description connects macroscopic states with possible microscopic arrangements. Gas expanding into a newly available volume has more positional arrangements. The informal word “disorder” can help intuition, but it does not replace a specified physical system and a careful account of its possible states.
References: [2]
Energy availability and work
Energy conservation does not determine how much work a process can deliver. The second law excludes a cyclic engine whose sole effect is converting heat from a single reservoir entirely into work. Entropy accounting supplements energy balances when analysing engines, refrigeration and irreversible changes.
References: [3]