Why is Arthur is in conflict with Lancelot?

The main conflict that exists between Arthur and Lancelot is because Lancelot is in love with Guinevere, Arthur's wife. This is made slightly more complicated by the fact that Arthur is the King and Guinevere the Queen, so technically Arthur can mete out justice if he determines Lancelot has broken the law. But the main reason this is complicated and painful is that Arthur and Lancelot care deeply for one another. Arthur deeply respects Lancelot's skill as a knight and warrior, all the more because when they first met Lancelot bested him in battle. Lancelot blocked Arthur's right of way, and Arthur challenged him to a fight. He decided to use his magical sword Excalibur, and when the sword broke on Lancelot's shield, he realized Lancelot was fated to win, and his use of Excalibur was arrogant and unfair. He throws the broken sword into the lake, and the Lady of the Lake returns it to him. This is an epiphany for Arthur, when he realizes Lancelot's inherent good nature, and the two are bound by deep friendship.

Which, of course, makes things exceedingly difficult when Lancelot falls in love at first sight with Guinevere. He tries to resist acting upon his impulse, but Guinevere is also smitten, and the two commit adultery. Once this act is discovered by Arthur, the kingdom is in an uproar. Lancelot runs off, Guinevere enters a covent, and Arthur loses his ability to rule as king and begins to waste away. He is shamed because his wife fell in love with another man (and this recalls the circumstances of his own conception, which occurred when Igraine betrayed her own husband Gorlois with King Uther). But he is also saddened by Lancelot's betrayal. Arthur's grief divides the kingdom for years. 

But years later, when Mordred (Arthur's son by his own half-sister Morgause—conceived via deception just as Arthur was) tries to take over the kingdom and usurp the throne, Lancelot returns from his exile to help Arthur take back what belongs to him. They fight beside one another once again, and their friendship is restored one last time.