A few quick comments. First, watch the spellchecker. You've got "plain" where it should be "plane". Microsoft can't solve this kind of problem for you.
Another thing to watch is the pronoun references. You clearly know who is who, but the reader who is going to read this one time, won't.
They had talked for a while, and enjoyed it. Her boyfriend had just broken up with her. He had still been pining for his first love, who had left him. Perhaps it was just bound to happen. They had grown amazingly open with one another. One night, she admitted.
'Her first boyfriend is a 'he' and the next sentence is 'he'. Was the 'he' in the second sentence the first boyfriend, or James? I'll bet on James, but it isn't clear. And making your readers guess interrupts their train of thought and interest in the story.
And the 'One night, she admitted.' is really part of the next paragraph, when the focus shifts to Des. Des, by the way, is never really 'introduced' by name. I thinkm we should learn her name a few paragraphs closer to the beginning of the narrative.
As for scene and context, it's been more than a few years since people could wait at the gate at the airport for arriving passengers. I'd shift the scene to baggage claim or the security checkpoint for more realism. You could have him thinking and anticipating and imaging what she looked like while passed the nameless throngs of people in the concourses. You could play off the anonymity of their relationship that was about to become very tangible.
There's also a few places where you shift focus from one character to the other in the same paragraph. That's generally not done, again for ease of reading.
But you do have a nice vignette here, and one that I know many will enjoy reading. Work on a few of the mechanics and some proofreading and you'll do fine.
