Alcatraz Prison closed on this day in 1963. While there were several infamous people who served time there (e.g. Al Capone and George "Machine-Gun" Kelly), it wasn't intended as a space to hold high-profile criminals. The popular understanding of Alcatraz as a dark, terrifying institution also doesn't really fit. At its peak, the island housed just 275 inmates, less than one might expect given its reputation, and living conditions were better than those at many other prisons.
During the California Gold Rush, the U.S. Army fortified the island to protect the San Francisco Bay as that city's population swelled. By the end of the 1850s the island was hosting its first military prisoners. The army would continue to use the island for that purpose until Alcatraz was transferred to the Department of Justice in 1933.
It was then used as the "prison system's prison," where they'd plop down troublemaking inmates who it was thought needed a behavioral adjustment. Alcatraz offered a hyper-structured, but bare-bones lifestyle compared to other facilities. Basic needs were met, but any form of recreation (or even work) was a reward for good behavior. Once the person started following the desired routine, they'd be transferred back to where they were incarcerated before.
Far as we know, no prisoner escaped the island successfully, though 36 of them tried, with most being caught and some dying. It's possible to swim there from the coast, but the journey requires serious athleticism that none of the prisoners could have had.
Now the island is part of the Golden Gate National Recreation Area and a popular tourism spot, seeing over a million visitors a year.
For a more detailed view of how Alcatraz worked and who was held there, check out this article from the Federal Bureau of Prisons.