Example 1
On Dan's first trip to New York.
What happened on Dan's first trip to New York? We may never know unless we fix this sentence fragment. It's missing both a subject and a verb. So let's put one of each in here. How about this? On Dan's first trip to New York, he visited Ellis Island, the Empire State Building, and Times Square. Much better. On Dan's second trip to New York, he watched the local wildlife scurry around in the subway tunnels. Look, Ma: a rat!
Example 2
The dejected third baseman.
In this fragment, we have a subject, the third baseman, but we have no idea what he did because there's no verb. So let's add one: The dejected third baseman threw his glove on the ground and stamped on it like an angry baby. That's some terrible sportsmanship, but now it's a wonderfully complete sentence.
Example 3
Left the carton of orange juice on the counter.
This fragment is missing a subject. Who left the carton out on the counter? If we add a subject—Rosie left the carton of orange juice on the counter—then we have a complete sentence, and we know who owes us some fresh juice. We hate you, Rosie.