The Superman robots are robots that physically resemble Superman, including his powers. More commonly seen in the Silver Age, the Man of Steel had then many robots that would help him keep his identity secret, posing either as Superman or as Clark Kent. At first the robots were just dummies, which Superman moved at super speed and he would throw his voice to make it seem like they were talking. Later on they became moving speaking automatons, which could act as Superman himself and serve also as maintance crew for the Fortress of Solitude.
Originally, it was the Silver age Superboy who used a pack of five robots to keep his identity as Clark Kent a secret. The advanced AI from the robots allowed some of them to develop selfconscious. Among the most notable example of robots are: Adam Newman, Ajax, Powerman, Robot X-3, Robot Z and MacDuff. But most of the time, the robots were only background characters with no development. There were also Clark Kent and Supergirl versions of the robots for the same purposes.
The Superman robots had all the powers of Superman in lower levels (the bigger exception was Superman-X), but they were immune to Kryptonite poisoning.
In the modern age, the Superman robots would resurface rarely as invention of Professor Hamilton. His massive irruption would happen when, under the mental influence of Dominus, Superman decided that the Earth populace needed to be protected at all times. To that end, Superman (pushed by the mental powers of Dominus) created the robotic legion of Supermen as a superpowered police force intended to keep humanity safe -- even from itself. This legion of Super robots would mantain tasks sucha as prevent crime, disasters and keep perpetual vigilance in loved ones (as Lois Lane) and enemies (as Lex Luthor). After recovering from his mental possession, Superman had all of the Superman robots destroyed except for one named "Ned", presumiblely, the same one who watched over Lois Lane, who watches Krypto in the Fortress, and acts as guardian/playmate for the non-superintelligent, though superpowered, dog.
The rest of the robots Superman would have decommissioned and put in a storage in Pluto, until Batman reactivated and modified them as a robot JLA to stop Grodd. Depiste the plan, it was a failure. The plan gives the League time to return to Earth.
One other defective Superman robot was kept in STAR Labs until it was reactivated by Indigo and caused the deaths of Lilith Clay and Donna Troy, before being defeated by the Titans and Young Justice.
On Superman: Red Son, the Superman-Robots are actually brainwashed soviet citizens.
In All-Star Superman, more robotic looking Superman Robots assist Superman in the Fortress of Solitude. One of the Superman Robots, Superman Robot Seven, is hacked by Lex Luthor. When Superman manufactures an exogene serum that grants a human Kryptonian powers for twenty-four hours,Superman Robot Seven smuggles the formula to Luthor. Later, when Solaris the Tyrant Sun attacks the sun, Superman has the Robots shut down the Fortress of Solitude. One stays behind to guard it as the power turns off.. During the fight with Solaris, all Superman robots seen perish, including Superman Robot Seven, who is seeking redemption for betraying Superman.
On earth-16, the Superman Robots kept the order in the planet. This caused worldwide ennui and depression among superheros as they found their lives without purpose. Later, their programing was spoiled by Alexis Luthor.
On the Tangent Universe the stronghold of that world's Superman is protected by fearsome-looking insectoid robots.
Alan Moore's Supreme would homage the concept with the Suprematons.
The trope also had a homage in Kurt Busiek's Astro CIty with Atomicus and his Atomic duplicates.
Other media appearances
Superman: The Animated Series: A Superman-Robot appears when Superman was kidnapped by Darkseid.
Justice League: The episode "A Better World" stars several robots.
Legion of Superheroes: A robot resembling Cyborg Superman appears.
The All Star Superman animated adaptation also includes the presence of the Superman robots.