A sentence has a subject (who or what is doing the action)
A verb (what is going on?)
Additional words that show what is being acted on.

A sentence can be very short: Kelly laughed.
..... "Laughed" is an intransitive verb, meaning you don't show what is acted on.

Some sentences need additional words: Kelly lost her backpack.
..... "Lost" is a transitive verb, meaning it needs another noun to complete the thought

Sentences may have a lot of additional phrases added in to give additional information.

Kelly, who hadn't slept in two days, laughed at the incredibly stupid joke.

After leaving the house at five in the morning, Kelly lost her backpack, the one with the "Hey Kitty" stickers on it.

 

A sentence has to be a complete idea all by itself. So, even if the sentence has a subject and a verb, it could still be a fragment.

Because Kelly hadn't slept in two days.

After Kelly laughed.

Since Kelly lost her backpack.

None of these are complete sentences because the word added to the front of the sentence means it needs a follow up idea to make it complete by itself.

In Japan, during the last war and just before the armistice.

Some of the students working in Professor Espinoza's laboratory last semester.

They left their work behind.

In a brand new house overlooking the harbour.

The committee disbanded.

While I clean up the yard.

ABCheckers guide to complete sentences
Writing a complete sentence
Perdue OWL--Avoiding fragments