This is a very large question, and with no specifics as to what aspects of the play are significant to which aspects of the culture, and no specifics as to which culture is being asked about, I can only use my best judgement in saying what I believe is significant about the play on the broadest possible level. This is an examination of Romantic Love, in all its various forms. It was created as a wedding entertainment for a noble person. The bottom line is that everything in Nature exists in relationship, and the harmony and/or disharmony that exists in any relationship manifests in everything. When the female and male archetypes, Titania and Oberon in the play, are out of harmony, it is shown expressed in the trials and difficulties between the lovers in the play. The play, then, is a working-out of these difficulties, leading to an ending of restored harmony, putting the Universe back in balance as a result. This is Shakespeare's gift to his patron and to us. The significance to the culture I am familiar with is the belief it is based upon, which says that for the universe to find itself in order and functioning properly, it is imperative for the male principle and the female principle to work in harmony, and when this is not the case, nothing can go right.
A Midsummer Night's Dream by William Shakespeare is a comedy that was written to entertain people. The fact that it continues to be produced, in both stage and film versions, and reprinted in book form suggests that many people continue to enjoy it and find it significant.
Next, A Midsummer Night's Dream is part of the English literary canon. This means that it is part of the shared cultural heritage of the Anglosphere, taught in schools and universities. What makes a literary canon important is that it give a common set of metaphors and cultural referents that English-speakers can use to communicate complex ideas. Although mass culture now has usurped some of this function of high culture, it is fairly inefficient at this task, as much of it is so ephemeral that it fails to communicate across generations; while people in their teens or 40s or 80s will all have heard of Shakespeare's plays, few of us remember the movies or TV shows favored by our parents, much less our grandparents.
The specific contexts in which A Midsummer Night's Dream is usually invoked are with regard to its comments on the nature of fantasy versus reality and the relationship of artistic genius to madness.