Nick Bottom is an important character because all throughout the play he unifies fantasy with reality, thereby uniting two of Shakespeare's major themes in the play and proving Shakespeare's point that there really is not a significant difference between fantasy and reality.
We first see Bottom unify fantasy and reality by first being the one who holds the mechanicals' play within the play together. Bottom is chosen as Pyramus, the lead character, because he is apparently the best looking of the group, has the best manners, and has the best voice. We learn about Bottom's looks and manners in Quince's lines:
You can play no part but Pyramus; for Pyramus is a sweet-faced man; a proper man, as one shall see in a summer's day; a lovely gentleman-like man. (I.ii.77-79)
We also learn later that Bottom speaks well, or pronounces well, when Quince mourns Bottom's absence, saying that they cannot possibly perform the play without him as Bottom is the very model of a "sweet voice," or well spoken performer, which is what Quince is trying to say when he says, "[H]e is a very paramour for a sweet voice" (IV.ii.11-12). The mechanicals' play represents a break from the illusion of reality that Shakespeare's own play creates. In that sense, the mechanicals' play represents fantasy within reality. Since Bottom is the character that holds the play within the play together and ensures its enactment, he unites fantasy, which is the mechanicals' play, with reality, which is the reality of the Athenian civilization inside Shakespeare's own play.
Another way in which we see Bottom unite fantasy with reality is that he is the only character who unites the fairy world with the human world. While the fairies interact with the human world, such as Oberon and Puck manipulating Lysander and Demetrius into falling in love with whom they should be mated with and blessing all the lovers in Theseus's house, the fairies ultimately remain separate from the human world. In fact the fairies and humans are so separated that the humans wake up disoriented, half believing what they experienced was real and half believing it was a dream. While Bottom wakes up with the same amount of disorientation, Bottom is the only character who directly interacts with the fairy world. He directly interacts with the fairy world by becoming Titania's lover and being waited on hand and foot by fairies. It is through this interaction that Bottom also serves to unite the world of fantasy with the world of reality in the play.
Hence, we see that what Bottom does throughout the play is unite two of Shakespeare's major themes, fantasy and reality, showing us that the dividing line between the two is actually very thin. Bottom serves to prove Shakespeare's point that there really is not a very significant difference between what one perceives is an illusion and what is actually real.