First appearance: Uncanny X-Men #185
Last appearance: X-Men (2nd series) #34
Description
Little is known about the “The State Home for Foundlings“ in Nebraska, save that the Summers brothers were brought there after their parents were kidnapped by the Shi’Ar and that Mister Sinister had a secret headquarter in the building’s basement.
Orphanage related plots
Years ago, the plane of the Summers family was attacked by the Shi‘Ar, and the engines caught fire. Finding only one parachute Alex and Scott's parents pushed the boys out of the plane, hoping that that at least they would survive. Even though the parachute was damaged by burning debris, the boys were saved thanks to Scott's mutant power manifesting, and his optic beam slowed their descent. However Scott cracked his head, and this caused a slight amount of brain damage, effecting the portion of his brain which controlled his latent optic blasts. Alex and Scott were brought to an orphanage in Omaha, Nebraska, secretly run by Mr. Sinister, who had registered Scott manifesting his mutant power. Sinister believed that Scott was the Summers brother with the most potential, so he had Alex adopted away from the orphanage to both separate them and render Scott emotionally vulnerable. [Uncanny X-Men #156, X-Factor #115, #39]
Over the years Scott was manipulated by Mr. Sinister, who posed as the institute’s director and also as his roommate Nate, who was his best friend. Sinister tried his best to keep Scott isolated from the other children and also faked his records about having severe brain damage, eliminating any chance to get adopted. Things seemed to change with the arrival of Dr. Robin Hanover. She cared for Scott and tried to get him to open up. Finally she succeeded when she found out that Scott loved planes and pilots. She even managed to find a foster family for him, but Sinister had them killed and Hanover was brainwashed. Scott knew that something strange was going on and he decided to run away as soon as possible. [Classic X-Men #41-42]
Sinister performed various tests on Scott and through trial and error he found out that only special ruby quartz glasses could block the optic beams. Sinister placed mental blocks on the boy so that he would not get exposed. [X-Factor #39] Needless to say, Scott did indeed turn his back on the orphanage a few months / years afterwards, only to be exploited the villain Jack O’Diamonds, till finally Professor Xavier invited him to join the X-Men.
When Sinister learned of the existence of a red haired girl with telepathic powers, he planned to kill her parents and have the girl taken to the orphanage too. 
However Xavier got to Jean Grey before. Sinister acquired a blood and tissue sample and grew a clone of her in the labs of his basement headquarter. He waited years for the woman to develop any powers and already thought he had failed, when one night a portion of Phoenix Force entered her body. He didn’t know that Phoenix tried to return Jean Grey the aspects of herself it had stolen, but Jean (lying in suspended animation at the bottom of Jamaica Bay) had rejected the images of death and destruction. Keyed to the genetic code of Jean Grey the Phoenix portion circled the globe until it had found a host in the clone, which Sinister named Madelyne Pryor. [Uncanny X-Men #241, X-Factor #38]
After the baby son of Cyclops and Madelyne was missing for quite a while, readers learned that little Nathan Christopher was in the care of Mr. Sinister. At an unrevealed location, that 
would later turn out to be the orphanage‘s basement, Mr. Sinister looked over a containment chamber which has the baby in it. The Limbo demon N’astirh watched from the shadows and having learned the child’s whereabouts he finally was able to bargain with Maddie. [Uncanny X-Men #239]
Following some clues from Destiny’s precognitive visions, Cyclops and Jean Grey went to the orphanage to search for Scott's son. They entered via the roof, so not to attract attention to themselves, however for some reason no-one recognized them in the building. Only a small girl with telepathic powers sensed them and Jean worked out that something in the basement was stopping the staff and children from noticing them. Cyclops remembered when he was at the orphanage and that he was often beaten by a gang who was led by a boy called Nathan. He recalled a memory that maybe Nathan controlled the staff 
as well, but he was unsure of the memory. As they went down to the basement in an elevator shaft, Nanny arrived with the Orphanmaker. They had come to recruit some of the mutant children in the orphanage. In the basement, Scott and Jean found dozens of babies kept in frozen pods asleep, among them little Christopher. A fight started between Nanny and the two heroes and in the confusion, Limbo demons stole the baby and headed to New York, Jean and Scott hot on their trail. [X-Factor #35]
N'astirh took Madelyne to the orphanage to show her where they found the child. Sinister showed up and introduced himself as Madelyne’s father or creator. Learning her origin Madelyne nearly went mad, and when Nastirh’s minions handed her the baby, she agreed to sacrifice him and establish Limbo on earth. By the baby’s death she would have revenge on the two men that caused her life of pain and misery - Cyclops and Sinister. [Uncanny X-Men #240-241] The rest of the Inferno plot was set in New York.
Years later, the Beast led a small team of X-Men to the by then abandoned orphanage in an attempt to obtain whatever data Mr. Sinister had stored about the Legacy Virus. They found the access to the secret underground complex and ran into a clone of the Marauder Riptide. After defeating him, the X-Men met Threnody, one of Sinister’s servants, more or less by choice. Yet she had no ill feelings for the X-Men and showed them Sinister’s huge genetic storage bank, possibly holding DNA samples of any mutant ever. The material was considered too dangerous and Threnody helped to blow up the base, yet remained at the complex to continue to work for Sinister once he returned. She claimed that she was using him as much as Sinister exploited her. [X-Men (2nd series) #34]
Later on it was discovered that all of Sinister’s bases are actually one and the same, connected to various points on earth through Tesseract technology. All the X-Men had accomplished that day was to destroy one of many entrances to Sinister’s headquarters. [Gambit (3rd series) #9]
With the access to Sinister’s base destroyed and the orphanage no longer in use, the place probably won’t get mentioned ever again.