It was used to locate people deemed guilty of crimes who had escaped punishment. After finding them near the end of their respective timelines to avoid changing history, the crew would administer their own punishment, usually in the form of some non-lethal, painful torture. Afterwards they could die in the manner recorded by history.
The Teselecta had a crew of 421 people led by Captain Carter. They could change its appearance to that of any humanoid being. The Teselecta visited 1938 intending to punish Adolf Hitler but realized they had arrived too early in his time stream. At the same time that the Doctor's TARDIS crashed into Hitler's office (due to Mels shooting the time rotor) and knocked the Teselecta to the ground, inadvertently saving Hitler. The crew witnessed Mels' regeneration into her River Song incarnation and recognised her as the worst war criminal in the universe for killing the Doctor in the future.
Following River's departure from Hitler's office, the Teselecta followed, miniaturizing Amy Pond and Rory Williams and taking on the form of Amy. As Amy and Rory were innocent of any crime, they were given security privileges that protected them from the Teselecta antibodies. The Teselecta attempted to punish River before being interrupted by the Doctor. With the Doctor demanding information on his death and Amy a relative of River, Captain Carter granted Amy access to the Teselecta's records on the Doctor and the Teselecta informed the Doctor of the true nature of the Silence and about the Question. The Teselecta soon resumed torturing River, but Amy used the Doctor's sonic screwdriver to revoke the crew's security privileges. Under attack by the antibodies, the crew began shutting the robot down, including its weapons systems, but were forced to beam out to a ship in orbit when they couldn't shut down the Teselecta in time.
After the Doctor died, Amy turned to the Teselecta which was still partially active. With Amy's privileges as a relative, she had the Teselecta shapeshift into River Song to show Melody that she and River were one and the same. Following his subsequent resurrection, the Doctor extracted data from the Teselecta databanks on his death at Lake Silencio. (TV: Let's Kill Hitler)
The Teselecta was later recovered by its crew. As the Doctor's death approached, the Doctor tracked it down posing as Father Gideon Vandaleur, a deceased envoy for the Silence. The Doctor was able to disable the Teselecta once again and promised to fix it once he got what he wanted. Having realised that the Teselecta was investigating the Silence, the Doctor asked for information on the Silence's weak link. The Teselecta crew pointed the Doctor towards Gantok.
As the Doctor prepared to go to his death, he visited the Teselecta to request that the crew deliver his invitations to his friends so that the Doctor wouldn't risk crossing his own timestream. Captain Carter promised to do so and insisted that no mater what the Doctor thought of them, the Teselecta and her crew were also champions of justice. After Carter asked if there was nothing more that they could do, the Doctor came up with the plan to have the Teselecta take on his form to fool everyone into thinking that he had died.
After River Song stopped herself from killing the Doctor, it fractured a fixed point in time and created River Song's World. In the alternate timeline, the Doctor remained in control of the Teselecta, changing its outer form to make it appear as if he aged to avoid suspicion. Eventually, the Doctor married River Song and revealed to River in the process that both he and the TARDIS were safely inside the Teselecta. Because she and the Doctor were the opposite poles of the disruption in time, River and "the Doctor" kissed, restoring the universe. (TV: The Wedding of River Song)
In the restored timeline, River blasted the Teselecta in the Doctor's form with her spacesuit's weapons pack, "killing" him. The Teselecta was set on a boat that was set ablaze and pushed out into Lake Silencio to burn. (TV: The Impossible Astronaut, The Wedding of River Song)
The Doctor later revealed his deception with the Teselecta to Dorium Maldovar while River revealed the truth to Amy and Rory. According to the Doctor, he "barely got singed" when the Teselecta was set ablaze in the boat. As a result, all but River, Amy, Rory, Dorium and the Teselecta crew came to believe that the Doctor was dead, the true fixed point in time that was recorded in history. (TV: The Wedding of River Song)
The Teselecta could resemble anyone or anything whose likeness it scanned. During its time in Berlin, 1938, it took on the forms of a cleaner, a Nazi official, a Nazi soldier (with and without a motorbike), as well as Amy Pond and River Song. A member of the crew said it had once taken the form of Rasputin, but with green skin. (TV: Let's Kill Hitler)
Though several of the Teselecta's features had been shut down before the crew were forced to abandon it, the shapeshifting aspect was still active and Amy was able to get it to take on the form of River Song by simply ordering the robot to access River's records and show River to her. (TV: Let's Kill Hitler)
The Teselecta was equipped with a miniaturisation ray, which it used to shrink people and transport them into the Teselecta. Compression fields were used to keep people miniaturised. (TV: Let's Kill Hitler)
If the crew's privileges were disabled, the antibodies would turn on the crew themselves and attack. The antibodies could presumably be disabled if the entire Teselecta was shut down as the crew attempted to shut the Teselecta down completely when such an event happened. However, they were not able to complete the task and were forced to call for outside teleportation to escape.
The loss of privileges keeping the antibodies from attacking did not appear to affect a person's control of the Teselecta's other features. The crew were still able to access its systems to attempt to shut it down and Amy Pond retained the access Captain Carter granted her to the Teselecta's databanks. (TV: Let's Kill Hitler)
Offensive and defensive features
The Teselecta was equipped with a non-lethal energy weapon, used within "justice mode", designed to torture its victims. It otherwise had no known built-in weapons, although targets could be miniaturised and finished off by the antibodies.
It was strong enough to withstand and take no significant damage from a collision from the Doctor's TARDIS at high speed. Handguns had very little effect on it. It had the ability to perform a "perfect faint" to plausibly avoid conversations with others. (TV: Let's Kill Hitler)
It was also fireproof, or at least highly fire-resistant; when the Teselecta posed as the Doctor and was burned after faking the Doctor's death, the Doctor, who was really inside, was unharmed, remarking after that the Teselecta was "barely singed". (TV: The Wedding of River Song)
The Teselecta possessed extensive databanks on individuals, organizations and events throughout time and space. Relatives of those targeted had certain privileges that allowed them to be granted access to the databanks by the Captain, but use of the databanks by civilians was generally forbidden due to the danger of knowing too much about one's own future.
In 1938, Amy Pond was granted access to the Teselecta databanks as a relative of River Song who the Teselecta was targeting. Once Amy requested access to the records on the Doctor, the Teselecta provided the Doctor with the information he requested by verbalizing it in a robotic manner. These databanks had information on the Doctor's death at Lake Silencio and that it was a fixed point in time. The databanks were able to provide information on the Silence and the Question, though not what the Question actually was as the databanks lacked that information. The Doctor was later able to extract the biographical data the Teselecta had on the time, date and location of his death.
After most of the Teslecta's systems were shut down, its databanks remained active and Amy, who retained her privileges as a relative, was able to access the records on River Song by simply speaking to the Teselecta. The Teselecta was then able to shapeshift into the form of River from its databanks at Amy's request. (TV: Let's Kill Hitler)