Today when we use that term I think we typically mean, "one of the main, more reliable, systems" we have at our disposal. It would typically run our "mission critical" software. I would say that the critical components of a semi-contemporary "mainframe" is that it supports a hypervisor, and is super reliable proably thanks to supporting redundant and hot swappable components. The trick is that today, that definition might be supported by not only all our cloud providers, but also by a good desktop workstation.
What makes/made them great?
Reliability, storage, and flexibility. Historically, such a systems put an enterprise's computing power into the hands of any employee with a "dumb VT100 terminal" (a glorified tv).
What can we learn from them?
System design tends to swing like a pendulum back and forth with focus alternatively on:
powerful centralized processing
or
autonomous distributed processing
At the moment, our cloud based platforms provide powerful centralized processing and we have increasingly seen moves to centralize on these platforms. The pendulum is reaching the end of the swing though as the end typically comes when architects envision and design systems that sit entirely on the "mainframe" and are accessed only though dumb terminals. Think todays efforts to bring "cloud gaming" to your phone or TV without a console or the myriad of system providers now attempting to sell your organization cloud based desktops.
There is no coincidence that we now see the rise of new generations of personal GPUs, CPUs and gaming consoles that provide every person what would have been a super computer not so long ago.
Just my 2 cents.
The reason it’s interesting to me is that we as a community have (it seems to me) split off from that world and use systems that were designed on shitty cheap mini computers that weren’t as good as mainframes (for some definition of good as).
You'll learn how to set up a z13 mainframe in your basement, how to use an emulator (Hercules) with open source MVS on your Linux/Mac/Windows machine, where to find the manuals. Very engaging and fun overall.
Back to your question: it's a whole another world. Mainframes are reliable, expensive as h&ll, thoroughly backward compatible, virtualized before anybody else, and still the invisible mainstay of our economy and whatnot (banks, insurance companies, DoD, airlines).
Its possible we have big, cheap I/O to play with now, these "bitcoin miner" motherboards that have been coming out may be the seed of another shift in computing.