Okay, you can argue that most incomplete sentences are just reduced forms. You can certainly go there. For example, when you say, "a hyperactive iguana," you can argue that it's just a reduced form of "A hyperactive iguana is chasing me." We don't know your life.
But that's the thing: we don't know your life. So when you write incomplete (or reduced-form) sentences that make sense in a certain context, just make sure to actually provide that context...if (and only if) incomplete sentences are stylistically acceptable. By that, we mean that these usually don't belong in an academic essay. Just sayin'.
In this section, we'll focus on situations that generally don't need as much context.
The most common subject-less sentence is the imperative sentence. We're talking "Run!" "Go!" "Stop!" and the like. Why are they okay even though they're just verbs with no subjects? Because they have an implied subject: you.
Yep, you got it. Interrogative words (like who, what, where, when, why, how) are reduced-form sentences because the subject and verb are implied. "What?" usually = "What (is that)?" or "What (did you say)?" Implication goes a long way.
"(The implied situation is) Adjective!"
Yep, that's a reduced-form sentence. It's technically a declarative sentence because, once again, we're implying the subject, and the verb is pretty much always "be." This sentence is basically a reduced form of "That's perfect." or "This is perfect." or some variation of these. Sure, we can't really know the subject out of context, but no matter when you say it, there's always one implied.
"Here (is an implied object)."
When you hand something to someone, you might say, "Here." No subject. No verb. But…still a sentence? Yep. Well, sort of. It's really a reduced form of "Here (is an implied object)." If you're holding out a slinky as you say it, it's implied that you mean, "Here's a slinky. (Take it, maybe.)"
Exclamations are the most primal expressions ever. We love 'em. Exclamations are usually just when you see or think of something and name it excitedly. You might see Mort the pug crossing the street, turn to your boyfriend, and say "Mort!" What you meant was "There's Mort!" but you were so excited, you couldn't even get the whole sentence out. Since there and is are implied, just the pug's name counts as a sentence on its own.
Or maybe you just got turned down for a fancy job. Ouch. You fall to your knees, dramatically look up towards the sky, and angrily spew, "Soulless big business conglomerates!" as everything around you spins out of control. (Hey, a dramatic exclamation calls for a dramatic situation.) In this case, nothing's really reduced; you just said what you said to express your distaste towards it. Kind of like how Malfoy disgustedly mutters "Potter" under his breath.
But there are also exclamations that are literally just used as exclamations, like "Huzzah!" and "Yay!" and "Wow!" and "Yikes." These words don't really have a semantic meaning (you can't describe a yikes), but they're used to express certain emotions.
Crash! Boom! Click! Squish! Pop! These are words that attempt to imitate the sound that they're describing. They're really not sentences in the linguistic sense, but we'll throw 'em here for good measure.