Sentence fragment
Generally, a complete sentence requires a subject and a complete verb phrase. A phrase with anything less is likely to be a sentence fragment:
I pulled up to the drive-through speaker. Where I placed my order.
Problem and cure: The first clause ("I pulled . . .") is well formed and acceptable. The second is a sentence fragment because it begins with a subordinating word ("Where"). To correct this, one would simply substitute the period with a comma and give "Where" a lower case "w."
Like the comma splice and fused sentence, a fragment is for many writing instructors a particularly serious offense.
It has become something of a cliché to begin a personal essay with a series of fragments:
Golden sand. The peaceful roar of the tide. A glimmering sun overhead. For me, Hawaii is paradise.
It is best to completely avoid fragments, though some practiced writers occasionally employ them for effect, as in the following:
Clem was handsome, rich, and smart. And the biggest jerk on the planet.
Technically, "and" (and "but," "or," and so on) is a subordinating word that, if used at the beginning of a sentence, creates a fragment, but everything depends on context. It is important for writers to know what their audiences expect--how much of a stickler their readers are for a close observation of usage rules.