I absolutely agree with kwoo1312, but I would add that Lodovico is also critical to the plot because he is a representative of the Duke. Earlier in the play, the Duke of Venice sided with Othello, even against a nobleman (Desdemona's father). As long as the Duke is firmly in Othello's corner, his position is secure. However, if the Duke turns on Othello, then he is far more vulnerable. If Iago wants to totally destroy Othello, he has to undermine that respect. If Othello had not killed Desdemona and they had all gone home to Venice, Lodovico would have carried word to the Duke that Othello had lost all honor and become abusive. That would have been very damaging to Othello.
So, in the play itself, he plays very little role in the schemes. He is the one who finds Roderigo injured and accepts Iago's explanations, but he doesn't advance Iago's plans much at all. However, he could potentially do Othello a lot of damage back home.
Lodovico is a nobleman who has come to deliver letters to Othello with his orders. When Othello sees the orders and sees that Cassio has been appointed in his place, he becomes so enraged that he strikes Desdemona after she is pleased about Cassio's new position. Lodovico witnesses this and is shocked and upset that Othello would strike Desdemona so violently for no apparent reason; therefore, Iago has made him a part of his plan by allowing Lodovico to come witness Othello's jealousy.