The following contains major spoilers for Avengers: Infinity War. Feel free to come back after you've seen it.
Marvel's Kevin Feige is a huge Star Wars fan. For that reason, many Marvel movies contain a secret tribute to the galaxy far, far, away. As the Star Wars saga contains several characters who have their hands cut off over the course of the first two trilogies, so too do many movies in the Marvel Cinematic Universe. Avengers: Infinity War, as it does with most things in the MCU, takes this to another level, as it actually has two characters that have their limbs chopped.
The first comes in the first major action sequence of the movie. Two of Thanos' Black Order have arrived on Earth to take possession of the Time Stone from Doctor Strange. Ebony Maw and Cull Obsidian then get into a major fight in New York that includes Strange, Wong, Iron Man, Spider-Man, and Bruce Banner. Obsidian, being a massive creature, becomes difficult for the heroes to overcome, leading to Wong opening a portal that dumps the villain in the arctic or someplace equally snowy. Obsidian attempts to jump back through the portal, but it closes before he makes it, only part of one hand makes it through, causing one massive hand to fall, severed by the portal, at the feel of Bruce Banner and Wong.
The second arm chop comes as part of what is, in the opinion of this writer, the single most badass moment of Avengers: Infinity War. Thor has arrived at Nidavellir with Rocket and Groot in order to obtain a weapon capable of destroying Thanos. Eitri the dwarf has successfully forged the ax Stormbreaker, but without a handle. Speed is of the essence, as Thor has been seriously wounded in creating the weapon and needs its power to survive. In a beautiful moment of self-sacrifice, teenage Groot grabs hold of the two halves of the ax with his own hand, brings them together, extends his arm, and then breaks it off himself, creating the handle of the epic weapon using his own body. Of course, Groot's injury isn't permanent (neither is the one to Cull Obsidian, we learn later), as he can simply grow a new hand, but it doesn't make the moment any less awesome.
These two moments follow a series of lost limbs throughout the MCU. Most famously, the Winter Soldier lost one arm when Bucky Barnes nearly died during World War II, but Thor also loses a hand during Thor: The Dark World and the Mandarin gets his arm chopped off in Iron Man 3.
The act of characters losing arms was mostly limited to Phase Two of the MCU, as a somewhat intentional reference to the second entry in the original Star Wars trilogy. However, Star Wars: The Empire Strikes Back certainly has a lot of tonal and thematic similarities to Avengers: Infinity War, so it makes some sense that the shout out was brought back.
Avengers: Infinity War is in theaters now.