Henry Lawson is the father of Miranda Lawson, a friend and (former) shipmate of Commander Shepard, and her genetic twin, Oriana Lawson. He works with Cerberus and answers directly to the Illusive Man. He runs Sanctuary on the planet Horizon during the war with the Reapers, luring in civilians to use as test subjects to find out Reaper indoctrination works and to find a way to control the Reapers.
Henry Lawson is one of the most powerful and wealthy businessmen in the galaxy and is obsessed with creating a genetic legacy to leave his mark on the galaxy. To this end, he creates children, attempting to give them the most desirable traits possible through genetics, taking his DNA and doubling his chromosome to create them. Through this he created Miranda Lawson. However, she was not the first that he made, only the first one that he actually kept, with the second being her sister Oriana Lawson. He conditioned Miranda to be perfect and controlled every aspect of her life leaving her with barely any friends or free will.
Miranda, wanting her sister to have a normal life away from his influence, ran away with her while she was still an infant. Miranda went to the organization that Henry donated large sums of money to: the human supremacist terrorist organization, Cerberus. The Illusive Man, seeing how gifted Miranda was gave her protection from their father, and relocated her sister to a new family to keep her hidden. She eventually moved high up into the organization becoming one of The Illusive Man's most trusted advisors. Henry Lawson has been hunting them, particularly Oriana, ever since.
Mass Effect 2
In 2185, Henry found out where Oriana was: living on Illium with her adopted family family. He hired the Eclipse mercenary gang to kidnap her and bring her back. Meanwhile Miranda Lawson was serving as Commander Shepard's second-in-command on the Normandy SR2 working to destroy the Collectors, a common goal of Shepard's and Cerberus'. Miranda got info that he found out where her sister was and with Shepard's help, thwarted the attempted kidnapping, and Cerberus relocated her and her family again.
Mass Effect 3
By 2186, Miranda had either been killed in the suicide mission to the Collector base, or defected from Cerberus, leading Henry Lawson to renew his support.
The Reapers also arrived and began their genocidal war on galactic civilization. The Illusive Man felt that the best way to win the war was to find a way to control the Reapers. This would not only win the war but put humanity in power above all other species. For this, he assigned Henry Lawson to find the secret behind how the Reapers control husks, their indoctrination, and finally how to use that to control the Reaper fleet.
To do this, Henry started Sanctuary on the planet Horizon. Advertised as a safe haven away from the war, this drew millions of families from across the galaxy like lambs to the slaughter. Once they were processed, they were experimented on and turned into husks in order to study husks and Reaper indoctrination. They hoped to reverse the indoctrination and turn it on the Reapers themselves.
At this time, Henry also located and kidnapped Oriana hoping to, once again, secure his genetic legacy and leave his mark on the galaxy, through both his manufactured daughter and his study of the Reapers which he felt would cause him to be seen as "the savior of humanity".
Eventually the Reapers discovered what he was trying to do and attacked Sanctuary, turning it into a ghost town. Meanwhile Shepard, who was tracking Kai Leng who stopped at Sanctuary, arrived and discovered the true nature of the place. Miranda had also tracked him down, while attempting to locate and rescue her sister from him and started leaving warnings to any refugees coming in as to the true nature of Sanctuary. Henry Lawson lived through the Reaper attack and when confronted by Shepard held Oriana hostage, using her as a human shield.
If Miranda was killed, Shepard can talk Henry into letting Oriana go and let him go in exchange, or not. If the Commander can not talk him, then s/he will shoot Oriana in the leg to get a clear shot at Henry.
If Miranda is still alive, even if Shepard talks him into letting Oriana go, Miranda will still kill him, ensuring that she and her sister can live life free of his influence and that Sanctuary and his horrible experiments will be shut down permanently.