Manipulator arm

The manipulator arm was a long, thin, mechanical arm found on all Dalek models bar some Emperor and special weapons models. The manipulator arm was found in the right-hand socket in the Dalek midsection, with the gunstick on its left. Its function varied depending on its design and it could be swapped out for a tool more specific to the Dalek's current needs.

Plunger
The ends of the manipulator arms typically were shaped like a plunger. For this reason, Dalek technology often had circular controls which the Daleks could arm or disarm by rotating. The black tip at the end, which gave the arm its "plunger" shape, could alter its form slightly to better interact with specific objects, for example, finding the combination of electronically-locked doors. (TV: Dalek)

Even if the Dalek died, its arm could still operate machinery for a live person (TV: Remembrance of the Daleks). The arm might have had a telescopic feature to to extend its length. (TV: The Dalek Invasion of Earth, Dalek)

It could also serve as a weapon. Placed over the nose and mouth, the arm could generate suction to suffocate and kill a human and enough traction to crush a human skull or rip through sheet metal. (TV: Dalek, PROSE: I am a Dalek) The Tenth Doctor claimed to have seen one crush a brick with ease. (PROSE: Prisoner of the Daleks)

At least in the case of the Cult of Skaro, the arm could also scan brainwaves for information or act as a quick intelligence scan. (TV: Doomsday, Daleks in Manhattan)

Imperial Daleks had a more conical variant with a slotted rim specifically designed to operate controls. (TV: Remembrance of the Daleks)

Flame-Thrower
Referred to by Daleks as pyro-flames, these were small cylindrical pieces at the end of the arm which shot a stream of fire to clear swaths of forest. (TV: The Daleks' Master Plan) The Daleks later used other flame throwers, which they referred to as flame guns, to melt through ice when searching for "the Abomination". The ends resembled the Progenitor Daleks' eyestalkss. (COMIC: The Only Good Dalek)

Seismic detector (or perceptor)
A small device which looked like a compass suspended on gimbals, the seismic detector was employed to locate enemy time machines outside the vortex. It could also detect approaching life-forms or the paths of fleeing ones, even through sealed doors. (TV: The Chase)

Electrode unit
The electrode unit was a device which looked like a radar dish. It could operate electronic controls remotely. It was used in this manner to enter the Mechanoid City. (TV: The Chase)

Sieve
Kaled mutants were inserted into the Dalek shell during assembly by a Dalek fitted with a sieve instead of a plunger. (TV: The Power of the Daleks)

Syringe
One of the members of the Cult of Skaro had an alternative attachment - a syringe. The syringe was fastened directly to the end of the manipulator arm in place of the plunger. (TV: Daleks in Manhattan)

Blowtorch
To cut through doors, early Daleks used a long, thin instrument with two spheres and a clear disc set midway up. It isn't clear if this was held or directly attached to the arm. (TV: The Daleks) Assault Daleks had a claw and blowtorch combination. (TV: The Parting of the Ways, PROSE: Prisoner of the Daleks)

Daleks in the Vault in the Crucible had a diamond-shaped interface to work with machinery and technology within the Vault. They resembled eight clawed arms. (TV: Journey's End)

Cutting tool
The Daleks used a hot iron to cut through metal. On Spiridon they used one to cut through a thick door. (TV: Planet of the Daleks)

Alternative weapons
One alternative weapon was seen on two of the Emperor's Personal Guard. It seemed to be an upgraded Dalek gunstick, but significanty larger, and with a large sphere on the end. The other side still possessed a gun. Another of the Dalek Emperor's guards had an arm with forward-jutting silver protrusions on a small ball. (TV: The Parting of the Ways)

Dalek X's elite guards had no manipulator arms. They had double gunsticks. (PROSE: Prisoner of the Daleks)

Behind the scenes
One Battles in Time trading card features a Dalek with a buzz-saw attachment, but this is never seen anywhere else.