Their relationship is based on reputation. Cassio says his reputation is the immortal part of himself: basically his soul. Othello likewise wrests his reputation upon a magic handkerchief, a symbol of what he thinks is Desdemona's love. Both men set themselves up to fail by placing too much stock in their public status, leaving them open to exploitation by the villain Iago.
Here's how their relationship unfolds:
Othello asks Cassio, who has otherwise been untested in battle, to be his lieutenant, instead of Iago. This is one of the primary reasons Iago seeks vengeance on Othello. Othello and Cassio's relationship is good in Venice.
But, on the isle of Cyprus, Iago uses Cassio against Othello. He gets him drunk on the night of what should be the Othello's honeymoon. Iago baits Cassio into getting into a fight with Montano, the governor of the island, and when he is injured, Othello is angrily roused from his sleep. Cassio confesses, and Othello strips him of his rank.
Later, Cassio seeks Desdemona to get back into Othello's good graces. This is bad timing, as Iago plants the seed that Cassio and Desdemona are secretly having an affair. Othello is already jealous of Cassio because he is young and white, and when Othello finds Cassio with the magical handkerchief that he gave to Desdemona, he plans to kill them both. Iago tells Othello that he will kill Cassio if Othello will kill Desdemona. Instead, Iago uses Roderigo to fight Cassio.
Cassio is injured by Roderigo but not killed. Othello soon realizes that he has been duped by Iago and apologizes to Cassio. Interestingly, in the 1995 film adaptation of the play, Cassio hands Othello the dagger that will kill him, such is his esteem for reputation.