To add to the excellent response above that discusses the storm's significance as foreshadowing for coming tragic events, I'd mention that the storm also changes the trajectory of the play. Othello leaves Venice in the belief that he is going to fight a war; he has been appointed to command the defense against a Turkish invasion of Cyprus. The storm, however, destroys the Turkish fleet—meaning that the play's plot will not involve bloody battles and military strategy (as we might have expected), but rather celebration and more free time in which Iago can manipulate off-duty officers (Cassio) and the recently married general (Othello) whose military campaign has turned into more of a honeymoon. The destruction of the Turkish fleet means manipulation and subterfuge will take center stage—not battles or military matters. War has been averted.
In Othello, Iago informs the audience early in the play that he has ulterior motives in serving Othello and, as he says, "I am not what I am" (I.i.66). He persuades Roderigo, who is in love with Desdemona, to rouse Desdemona's father (Brabantio) to warn him that she has betrayed her father by running off with Othello. Brabantio is distraught and immediately sets out to track down Othello. The quarrel is only settled when Desdemona herself vouches for Othello and reminds her father that a woman will leave her father for her husband, such as his own wife did. However, Brabantio remains shocked by Desdemona's actions and even warns Othello that, as he says in Act I, scene iii, line 293, "She has deceived her father, and may thee." This foreshadows later events when it is Desdemona who is betrayed when Othello does not believe her. It is significant and ironic that, at the end of Act I, Othello, unaware that Iago is responsible for Brabantio's outburst and for casting doubt on Desdemona's integrity, leaves Desdemona in "honest" (294) Iago's care. Iago will then convince Roderigo that he still has a chance to win Desdemona, all the while planning his next move in destroying Othello. Act I closes with Iago plotting to involve Michael Cassio and to cast doubt on his honor. Iago knows he can manipulate Othello because Othello "thinks men honest that but seem to be so" (394), meaning that he is a trusting person and as long as a man has the outward appearance of being honest, that will satisfy Othello.
It is then fitting that Act II begins after a storm. The Venetians themselves arrive safely but the Turkish fleet has been lost at sea as the fleet is unable to withstand such a storm. Cassio arrives followed by Iago and it is as if the storm announces how almost indestructible Iago is while the audience still wonders about whether Othello will make it. The plot is advanced because the audience knows that Iago is plotting against Cassio and therefore the storm foreshadows the tragic events that will follow and Cassio's unwitting involvement in Iago's plan. No one and nothing is safe from Iago. The fact that Desdemona is with Iago confirms that Iago will get the better of everyone, even the storm. The storm signifies the conflict that will follow and drives the plot forward as Iago's schemes will begin to unfold.