I'll have a go at explaining. This is a super abridged version just to get the main details in.
They're not brothers, but they do behave in that kind of way.They grew up together on Galifray, went to school together, but then The Master goes mad after he looks into that time vortex.
Then there is a Time War. Both The Doctor and The Master escape while Galifray is destroyed.
Then as an adult, he goes to Earth and takes it over by brainwashing the public and becoming Prime minister of the UK. Meanwhile, The Doctor is looking for him, because they are the last time lords, so The Doctor feels personal responsibility for saving Earth and trying to help his old friend.
In the end, the Master is defeated. But instead of trying to kill The Master for what he did to Earth, The Doctor says he forgives him and wants to take him in the TARDIS and help him because they are the last of their kind. The Master, not wanting to accept this, lets himself die instead of regenerating his body. The Doctor mourns the loss of his old friend and then burns his body.
Later, The Master comes back, but his body is faulty. The Doctor takes it upon himself again to save Earth and The Master. It culminates in the return of the Time Lords, who used The Master to do this. Realising this, The Master sacrifices himself and saves Earth.