One faction down, one more to go. After the assault at Railroad HQ (see End of the Line), now is the time to take the battle to The Brotherhood of Steel.
After initiating with quest with Father, head to the Advanced Systems lab and speak with Doctor Watson (or Doctor Li if you didn't complete the From Within quest). The doctor, along with a very enthusiastic Rosalind Orman, will give you the mission details. It's an assault on the Brotherhood's ground base at Boston Airport, and you'll need to use a giant robot to take down the faction's flying base.
After you reach the airport, you need to destroy three generators, all of which are conveniently marked as waypoints. Taking out each one will open portals for The Institute's Synths, so it's in your best interest to finish this task as soon as possible.
After that's done, head over to Liberty Prime, the giant robot in the middle of the airport terminal. Take the stairs up to Prime's head and await the arrival of a Synth. Your job will be to protect the Synth until it manages to override the robot's controls. If the Brotherhood's defenses manage to destroy the Synth, don't worry. A replacement Synth will soon appear to finish the job.
Once Liberty Prime is out of the Brotherhood's control, evacuate the airport. As you reach the airport's perimeter, you'll automatically teleport to a safe distance to witness the Brotherhood's destruction from afar.