Iago, who is the villain of this tragedy by Shakespeare, and several other characters are jealous of the relationship between Othello and Desdemona, including Roderigo. Roderigo (and probably Iago, on some level) wants Desdemona for himself, so naturally, he is jealous. Iago is jealous of Othello, too, because he is a skilled general and is much more talented and strong than he is. Iago has little love or respect for his wife, Emilia, who is Desdemona's lady-in-waiting. He treats her cruelly and with disdain, so he is probably jealous of the true love Othello and Desdemona share.
Other characters, including Desdemona's father, disapprove of their relationship because of how quickly they married and because they did it in secret. Also, some are upset because Othello is a Moor, which is a different race than Desdemona.
Others try to be supportive of their relationship, including Cassio and Emilia. They, however, get sucked into Iago's evil plan to make Othello kill Desdemona, which ultimately succeeds. All of the characters are touched in some way by this tragedy.