The play is essentially about love. The conflict arises out of this.
First there is the impending marriage of Theseus to Hippolyta. Theseus tells her in I.1 that he conquered her into marriage. When he leaves the scene, he say, "Come, my Hippolyta." The use of the word my indicts possession.
Hermia's father demands the ancient privilege. Egeus is another man who looks at females as possessions. He wants to force his daughter to marry the man he has chosen, Demetrius instead of the man she loves, Lysander.
It is also revealed that Demetrius "made love to Nedar's daughter" who happens to be Helena. Made love is not what we think today but we could say that he was dating her.
So in I.1 we learn that Thesus won Hypolita, the Queen of the Amazons, in battle and despite his telling her he loves her, he also considers her a possession, the spoils of war. We also learn that Hermia loves Lysander and that until Egeus decided he wanted his daughter to marry Demetrius, he was in love with Helena.
When Lysander reveals his plan to Helena, she tells Demetrius the plan and follows him. As a result, the four young lovers end up in the forest that is ruled by night by Oberon, Titiana and the fairies, a magical place where anything can happen.....and does.
In I.3 we meet Oberon and Titania, King and Queen of the faries and the rest of fairyland. We discover that the powerful king and queen are having a domestic quarrel that affects not only fairyland but the "real" world. Their argument concerns a young boy. Oberon feels it is time for this page to become a squire. In other words it is time for the boy to go from the world of women (as a page) into the world of men. The references of page, squire and knight are all used in connection to this young boy. In order to get the boy away from Titania, he uses Puck and magic. He also decides to help out Helena and sends Puck. As a result both men end of "in love" with Helena. Also, Titania falls "in love" with Bottom whose head has been replaced by magic with the head of an ass. (This was an in-joke for the Elizabethan audiences since Bottom was played by Will Kemp who would always manage to work in the fact that his character was an ass. See Dogberry in Much Ado.)
The application of the love juice and the release of this juice provide the comedy.....both men pursuing Helena and Titania ennamered by an ass. Once Oberon has the boy, he releases the spell on Titania and commands Puck to make things right with Bottom and the four young lovers.
In the end, true love prevails and the not only do Oberon and Titania reunite in concord bringing the same to the rest of the world but the lovers are paired correctly and when Thesus has learned that Hypolita in not his possession but his equal when he says, "Come, Hypolita" in V.
The play put on by Bottom and company is a parody of love.
So, the conflict in the play is romantic love in its various forms.