There are a number of contexts in aviation where it's very important to verify that another person has heard and understood you, but it's often not a good time or place to have an extended conversation about the information that has been conveyed. When given a clearance on an IFR flight plan, the pilot reads back the entire clearance. It may be in slightly different words, but the words won't change much because they still have to comply with aviation terminology. The controller listens to the repeated clearance, called the readback and acknowledges or corrects it. If there is an error, the pilot has to read back the corrected portion until it is all worthy of the response, "readback correct."
When the flight is VFR, the pilot need only acknowledge the clearance, usually by repeating her callsign. So the controller says, "Alpha Bravo Charlie cleared direct SPOTT," and the pilot says "Alpha Brav Charlie." The message, "I hear and obey o great air traffic controller" is implied. There are exceptions both ways: sometimes in very busy airspace, like Oshkosh airshow arrivals, the controllers just talk and pilots acknowledge by obeying. Sometimes VFR pilots working in complex airspace where the bulk of the traffic is IFR will read back instructions. It's okay to do so any time the pilot wants to double check that what they heard is what the controller said.
Between pilots in the cockpit one may give the other information, information as simple as "Edmonton's on zulu, it's on the clipboard." That means that non-flying pilot listened to the recorded terminal information message on a dedicated frequency broadcast by the tower at Edmonton International Airport (insert sadness here as I miss the late Edmonton City Centre), wrote down the reported airport conditions, including the fact that the identifying letter for the latest recording is "Z" and presumably put the clipboard where the flying pilot could see it. The flying pilot will probably say "Check." Ah, half the time if the cockpit and radio are quiet she'll probably say "thanks" but the meaning is the same, roughly, "I acknowledge that, and I'll deal with it when I need to." It isn't really a favour one pilot does for the other, just both working as a team.
"Flap 15 selected and transiting ... approach flap set."
The word "roger" also means the same thing. It's an aviator's way of saying "Gotcha" or "Cool" or "OK." Another old word with a similar meaning is "Wilco", short for "will comply".
Here's what prompted this post. My dentist e-mails me with an instruction. She knows I'm a pilot, because somewhere in the regulations it says I'm supposed to let medical professionals who treat me know that, but the instruction has nothing whatsoever to do with aviation. I don't have to give her any information, just let her know I received the e-mail. I type "check" because it's really the first thing that springs to mind when I want to be perfectly clear about having received an instruction. That reminds me of a story though.
A student pilot was flying in a busy circuit and before she got a word in on the radio to report that she was on the downwind leg (ready for sequencing instructions) the controller pre-empted her call by saying, "Check you're downwind, number three" meaning, "I see you're on downwind, follow the two airplanes ahead of you." The student wasn't familiar with that use of check so interpreted it as a command "check your downwind," that is "do something to your downwind." The student was sure that she was flying in the right place at the right speed so didn't change anything, just asked her instructor about it afterwards. I'm the instructor in that story, and I didn't want my dentist to think I was telling her to check something, so I backspaced over it and started to type "wilco." But that's even more jargon-y. Roger? I ended up with "Sure thing!" Overthinking it perhaps but I offended a medical professional once with a terse automatic aviation-style reply, and I don't want to do that again.