Most good stories start with a fundamental list of ingredients: the initial situation, conflict, complication, climax, suspense, denouement, and conclusion. Great writers sometimes shake up the recipe and add some spice.
Initial Situation
The play opens in the living room after breakfast, and the first thing we hear is James insisting that Mary has gotten more plump and sassy again. James and Mary appear fairly content, and Jamie and Edmund seem to be laughing good-naturedly at their father.
Even if it only lasts a couple pages, the initial situation is a happy one. This isn't paradise, and we can sense that things haven't been (and won't be) perfect in the household. Nonetheless, O'Neill sets the scene with mirth.
Conflict
At the end of Act I though the end of Act II, Scene One, the problems that shape the play begin to reveal themselves. Mary has a mysterious but nasty habit, Edmund's sick, Jamie is a degenerate, and James is cheap.
Things head south quickly, and we see that the characters are dealing with personal and interpersonal conflicts.
Complication
As if they didn't have enough to deal with, the Tyrones start getting busy making things worse. Act II, Scene Two and Act III brings about the escalation of infighting. Mary's morphine addiction is back and worse than ever, Edmund has consumption, and the men in the family start drinking a bit too much.
Basically, the Tyrones' conflicts become clear and we see that they have very serious problems.
Climax
We've reached the end of Act IV. After binging on alcohol, the boys come home to find Mary high on morphine and hallucinating about her past. With no idea of what's going on around her, she's a shell of her adult self. Mary reverts to her much-lamented girlhood, when she felt that she had her whole life ahead of her. Jamie and James look on with weary resignation.
Everything comes to a head by the end of the play – the drug and alcohol users have hit bottom, and all the male Tyrones stage epic showdowns with one another. There may be hints of reconciliation among them, but Mary's sad situation takes center-stage.
Suspense
The future of the Tyrone family is uncertain.
Well, the curtains come down right on the climax, so we never really learn the aftermath (although, the epigraph could also work as an epilogue – see "What's Up with the Epigraph?" for more on this). Who lives? Who dies? Will Mary ever kick her habit? Will Jamie and James be consumed by their alcoholism? Will that sanatorium work out for Edmund? Can the brothers ever be friends? Guess we'll never know.
Denouement
O'Neill decided to end the play during the climax, so we don't really get a denouement.
No dice.
Conclusion
All the concluding is left for you. Congrats!
If you want, you can write the sequel: Long Night's Journey Into Day.