Non-living
Non-living and unliving are terms that refer to creatures that have never been alive. The term non-living was introduced in the The Shadow of Death manual, the only use of 'non-living' in the manuals (used to describe creatures unaffected by the Elixir of Life). 'Unliving' is not used at all in the manuals but replaces non-living in the in-game description of the Elixir of Life. The manual states that the artifact "does not work on Undead or non-living creatures". Additionally, the non-undead creatures not affected by the health bonus of the Elixir of Life, (golems, elementals and gargoyles), have other immunities in common, which allows them to be considered one group of 'non-living' creatures.
Stone and Obsidian Gargoyles are also considered non-living, but they are still affected by morale and mind spells.
Though war machines as units on the battlefield can be considered non-living in some sense, they have their own system of immunities.
Crystal Dragons are described as being "made entirely from red crystal and brought to life through magical means..." and are referred to as dragon golems in the campaign. Despite these references they don't have any features that would set them apart from any other living creatures.
Non-living creatures
Golems
The only common trait of golems among all other non-living creatures is spell damage resistance.
Elementals
Air and Storm Elementals
Water and Ice Elementals
Fire and Energy Elementals
Earth and Magma Elementals
Psychic and Magic Elementals
Gargoyles
Immunities
All non-living creatures are unaffected by the 25% health bonus of the Elixir of Life, by morale (except for gargoyles), and the following spells and abilities:
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Archangel: ● resurrection
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Mind spells (except for gargoyles)
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Unicorn: ● blind
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Fangarms: ● hypnotize 
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Creature abilities
Azure Dragons: ● fear
Ghost Dragons: ● aging
Lich and Power Lichs: ● death cloud
Mighty Gorgons: ● death stare
Pit Lords: ● summon demons
Vampire Lords: ● life drain
Wyvern Monarchs: ● poison
Sea Serpents and Haspids: ● poison 
Zombies: ● disease