FPGA is basically an integrated circuit (chip) that can be reprogrammed. It can be useful for prototyping new chip designs or for reverse-engineering and/or implementing existing chip designs. So chips like the 6502 or the 68000 can be programmed in to the FPGA, and it will then essentially do the work of that chip.
As it works on a hardware level, it means that the way it operates is very close to original hardware - even more so than emulation, and in a way that feels more authentic. Not only that, but the speed and accuracy is also much closer to the original hardware than emulation can offer.
The MiSTer is an Open Source project based on the Terasic DE10-Nano and as a bare-bones set-up this is all the hardware you need. At least for a couple of cores (a "core" is the code that programs/initialises the FPGA).
The MiSTer's strength is in it's open nature. There are quite a myriad of expansions you can get for it to extend its functionality, and many (software) projects to simulate different computer systems. For example, it's technically possible to have a MiSTER installed inside a (modified) Amiga 1200 or BBC Master and to have it operate almost seamlessly as though it was the original computer.
One thing to bear in mind is that the MiSTer's priority is on compatibility, and to ensure as close to 100% accurate replicas of original hardware as possible.
Vampire V4 Standalone
Vampire is a familiar name to anyone who has seen any Amiga-related news in the past few years. Apollo have released a series of highly-rated Vampire accelerator cards for Amigas. They can be fitted easily inside original hardware, without soldering, and provide all kinds of benefits including internet access and vastly improved speed and memory.
The V4 is an evolution of this, to the point where it's become what they are branding a "brand new Amiga". The idea is to bring compatibility of original Amiga software with some specifications which come much closer to what you might expect in modern hardware. They have developed what they are branding a 68080 a "code-compatible" processor in the 68000 family, and the unit is equipped with HDMI (supporting up to 720p resolution) and USB.
Note: while it has USB ports, they are not 100% compatible with all USB devices, and specific ports are allocated to mice, keyboards and joypads with a given list of compatible hardware.
Both of these projects are fantastic in very different ways. The MiSTer supports accurate simulation of original hardware in a world where the original hardware is 30 or more years old, but the Vampire V4 is attempting to bring a brand new Amiga (with as much backwards compatibility as possible).
They are also both in active development, and I believe both are worth keeping an eye on if you have any interest in retro gaming or old computer hardware. I would note that MiSTer is a completely open source project with a slightly bewildering array of support and applications, while the Vampire is much more focused.