Othello succumbs to Iago's assertions so easily because he is an emotionally weak man. A stronger man would ignore Iago and trust in Desdemona, or even take Desdemona's side of the story into account. Not Othello. He is consumed by an intense love for Desdemona and retains so much doubt that it leads to his blinding jealously and eventual tragic end.
We must remember this is a Shakespeare piece and people often become obsessed and controlled by emotions, such as love, in Shakespeare plays. These intense emotions have a tendency to weaken a character's otherwise better judgement. Knowing that, it makes sense to say that Othello loves Desdemona so much that even the idea that she is cheating on him is enough to set in motion a raging fit of jealousy and suspicion that is likely to end in tragedy. This makes Iago's job easy. By planting this idea in Othello's mind, along with the slightest bit of evidence, Othello will easily become obsessed with the idea that Desdemona is unfaithful to him. A man with a stronger hold on his emotions would instead talk to Desdemona and believe her when she says she is not cheating; a stronger man may not act as irrationally as Othello does.
Othello's race also plays a large role. Othello is black and although he is a highly respected man and decorated war hero, his race does make him an outsider in Italy. Even Desdemona's father didn't want her marrying a black man and accused him of using witchcraft to make her fall in love with him. The idea that Desdemona could be falling for a smooth talking, handsome white man definitely works towards his theory of Desdemona's affair and fuels his jealousy. Again, a man who had more faith in himself, a stronger man, would not succumb to this idea as easily.
To further illustrate this, imagine that you are Othello. Although you are a highly respected and decorated war hero, your race still makes you an outsider. You are head over heels in love with this beautiful Venetian woman, but you have to marry her in secret because her father doesn't like Africans. Even when he finds out, he accuses you of using witchcraft to make his daughter love you. Then your beautiful wife starts spending time with a very handsome and respected white man. Then Iago, someone you trust, tells you that your wife might be cheating on you and brings you some incriminating evidence. Taking all this into account, and the fact that you exist in a Shakespearean world and you are taken over by an intense sense of love that could drive you to kill, naturally you wouldn't be emotionally strong enough to deal with all of this rationally and would be blinded by jealousy. You would believe every word Iago says because you are not a very strong man when it comes to emotions, faith in yourself, and faith in the woman you love. You would become blinded by jealousy and be driven insane. This is precisely what happens to Othello.
The character of Othello in William Shakespeare's play "Othello" is one of the more complex and fascinating ones in the Shakespearean canon. Despite the excellent points of the answer above, I am not convinced that we should view Othello as weak. He is a military hero, renowned for his bravery. He is, however, less self-assured in matters of love and friendship than he is in miltary matters, something quite understandable as he is a professional soldier who has had little experience of or leisure for flirtation or even a social life. Moreover, as a Moor, he is well aware that he constantly needs to prove himself in Venetian society, and that many people will discriminate against him because of his race. Rather than being a sign of weakness, this is an example of the clearsightedness that makes him such a good military leader. Unlike Iago, who has a massive sense of entitlement and greatly overestimates himself, Othello is capable of self-criticism, something very few people are strong or secure enough to manage.
The greatest character flaw in Othello is that he is prone to jealousy, and that jealousy is his downfall, as Iago correctly (if with double-edged malice) warns:
O, beware, my lord, of jealousy;
It is the green-ey'd monster, which doth mock
The meat it feeds on ...
Although we can say that jealousy is Othello's weakness, as well as himself being so honest that he does not recognize Iago's duplicity, nonetheless he does not seem a weak character, but a strong character with flaws (is anyone completely without flaws?) that lead him to fall victim to the evil machinations of Iago.