The term covers magical clones, golems, homonculi and many others - things that are animated by magic alone usually don't qualify, as there needs to be some aspect of construction and snapping your fingers and having the table spring to life just doesn't make the grade. Acari on the other hand, may or may not qualify as well - depending on your definitions, they many not be artificial enough … the category does not usually include species created by "magical genetic engineering" so an actual species created by alchemy may also be disqualified. Indeed being capable of meeting the typical definition of a species by reproducing probably disqualifies something from being a construct at all - if constructs want kids they have to build them.
Of course, the whole concept of magical artificial intelligence crops up when constructs are involved - in some cases a construct runs off a written "program" (golems are quite often portrayed like this), whilst in others they require a living brain (or some kind of spirit) bound into them before they have any measure of control.
This, then, means that the boundary between constructs and undead may become blurred - in the Hollywood portrayal1, how does Frankenstein's Monster differ from a zombie stitched together from a variety of corpses? If the creature eats and breathes the answer may be simpler, but if it doesn't? In a similar way, does binding a dead person's spirit to a construct actually make it undead? Does what the construct is made of alter that?
For those who wish to transplant sci-fi tropes into fantasy, or invert Clarke's third law, constructs are very useful indeed.
Regardless, Frankenstein Syndrome is to be expected, plot wise.