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. When the Eleventh Doctor, Amy, Rory and Mels crashed the TARDIS into Adolf Hitler's office, the Teselecta was trying to punish Hitler, but had encountered him too early in his time stream. When Mels regenerated, the Teselecta crew recognised her as Melody Pond. They stopped their pursuit of Hitler to punish her instead, as their records showed her as the criminal who would kill the Doctor. They were thwarted by her parents, who turned off the devices protecting the crew from the ship's antibodies. They teleported to a spaceship above the Earth, abandoning the Teselecta. The Doctor then used the Teselecta to show Melody that she would become River Song, someone very close to the Doctor. This helped Melody to resist her childhood programming as the Silence's assassin. The Doctor also had it reveal the Silence's purpose and extracted data from it regarding his death. (TV: Let's Kill Hitler)
The Teselecta was later recovered by its crew. They investigated the Silence by posing as Gideon Vandaleur, a deceased envoy of that cult. The Doctor approached the Teselecta for information. After sending the Doctor to someone who could help, the Teselecta was again approached to deliver the Doctor's mail lest he cross his own timestream. The Captain asked the Doctor if there was anything else they could do, as they really did believe in justice and wanted to help, despite the Doctor's contempt. The Doctor asked for the Teselecta to pose as him and fake his death. (TV: The Wedding of River Song)
In the main timeline, River Song "killed" the Teselecta, in the guise of the Doctor, and it was "barely singed" when "the Doctor's body" was burned on a funeral pyre. (TV: The Impossible Astronaut,The Wedding of River Song) In another, she prevented the Doctor's "death". This created an alternate universe in which, now that a fixed point in time had changed, time was breaking down. It found this world's versions of Amy and Rory and went to Area 52 with them.
Still posing as the Doctor, the Teselecta and River performed a hasty wedding ceremony. The Doctor revealed to River that he was hiding inside the Teselecta, so she would allow time to be restored. When River and the Teselecta touched, the timeline restored itself and Time continued on its wanted course. (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)
Miniaturisation ray Edit
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)
Offensive and defensive features Edit
The Teselecta was equipped with a non-lethal energy weapon 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)