I just don't get it. Any manual is too technical. What are flat and segmented memory? Ways of addressing a memory, ways of organizing bytes in memory? Which of them is best for 32-bit computers? Can anybody explain it? What does real-mode and protected-mode have to do with flat or segmented memory? Thanks!
If you're only interested in applications running on existing 32/64 bits operating systems, you can simply forget segmented memory. On 32 bits OSes, you can assume that you have 4 GB of “flat” memory space. Flat means that you can manipulate addresses with 32 bits values and registers, as you would expect.
On 16 bits processors, I believe an address was 20 bits wide, and you couldn't store that in a register, so you had to store a base in one register, and to specify an actual address, you had to add an offset to that base. (If I remember correctly, the base was multiplied by 16, then the offset was added to get the actual address.) This means that you could only address 64 KB at once; memory had to be “segmented” in 64 KB blocks.
To be honest, I think the only reason beginners still hear about that is because a lot of old 16 bits tutorials and books are still around. It's really not needed to understand how a program works at the assembly level. Now if you want to learn OS development, that's another story. Since a PC starts up in 16 bits mode, you will need to learn at least enough to be able to activate the flat 32 bits mode.
Just noticed you also asked about real mode vs protected mode. Real mode is the mode that MS DOS used. Any program had access to any hardware feature, for example it was common to directly talk to the graphics card's controller to print something. It didn't cause any problem because it wasn't a multitasking OS.
But on any modern OS, normal programs don't access hardware directly, they don't even access the memory directly. The OS manages the hardware and decides which process gets to run on the processor(s). It also manages a virtual address space for every process. This kind of feature is available with protected mode, which I believe came with the 386, which was the first 32 bits processor for PC.