There are many different types of love represented in Shakespeare's A Midsummer Night's Dream, and certainly more than three. However, in this answer, I'll focus on the types of love that I find to be most prominent/important in the play: romantic love, unrequited love, and forced/fake love.
Romantic Love: This type of love is pretty straightforward, as it's the kind of love that occurs between people who really, genuinely, deeply care for each other. There are multiple examples of romantic love in the play, as it can be found pretty generally in most of the play's natural couples. Lysander and Hermia, Theseus and Hippolyta, and even Oberon and Titania all experience romantic love within their relationships (although, it's worth noting, this latter couple has a pretty weird way of expressing it at times). Essentially, all of these couples genuinely care and love one another in a romantic fashion.
Unrequited Love: There are less examples of unrequited love, but it's still important. At the beginning of the play, Helena has unrequited love for Demetrius, as she loves him dearly but is repeatedly scorned by him. As such, her love is not returned by the object of her affections. However, Demetrius ultimately does return Helena's affections by the end of the play, but that doesn't diminish the importance of her original, lonely expressions of love.
Forced/Fake Love: This kind of love is obviously artificial, and it serves as one of the play's primary comedic devices. Lysander, Demetrius, and Titania are all given a potion that causes them to falsely love the first thing they see upon waking. As you can imagine, this causes quite a lot of confusion and hilarity, as it motivates Lysander and Demetrius to fight over Helena, and also makes Titania dote upon the bumbling Bottom. However, while funny, the forced/fake love motif has sinister undertones. Forcing one person to love another is potentially traumatic and, though it's funny within the play, it certainly wouldn't be very funny in real life. As such, though forced/fake love makes us laugh while watching the play, it should also cause us to think more critically about the boundaries in relationships that cannot be crossed.