Memory-mapped I/O and port-mapped I/O are two complementary methods for I/O.

Memory-Mapped I/O

In memory-mapped systems, the I/O device is accessed like it is a part of the memory. Load and Store commands are executed for reading from and writing to I/O devices, just like they are used for the memory (port-mapped has special commands for I/O). This means I/O devices use the same address bus as memory, meaning that CPU can refer to memory or the I/O device based on the value of the address. This approach requires isolation in the address space: that is, addresses reserved for I/O should not be available to physical memory.

Below is an image of a simple, basic computer system. The case is much more complicated in contemporary systems.

Port-Mapped I/O

According to Wikipedia

Port-mapped I/O often uses a special class of CPU instructions specifically for performing I/O. This is found on Intel microprocessors, with the IN and OUT instructions. These instructions can read and write one to four bytes (outb, outw, outl) to an I/O device. I/O devices have a separate address space from general memory, either accomplished by an extra "I/O" pin on the CPU's physical interface, or an entire bus dedicated to I/O. Because the address space for I/O is isolated from that for main memory, this is sometimes referred to as isolated I/O.

As for the advantages and disadvantages: since the peripheral devices are slower than the memory, sharing data and address buses may slow the memory access. On the other hand, by the I/O simplicity memory-mapped systems provide, CPU requires less internal logic and this helps for faster, cheaper, less power consuming CPUs to be implemented. The logic is similar to that of RISC systems: reduce the complexity, get a more dedicated and a robust system which comes quite handy for embedded systems, for example.

On the contrary (again from Wiki):

Port-mapped I/O instructions are often very limited, often providing only for simple load and store operations between CPU registers and I/O ports, so that, for example, to add a constant to a port-mapped device register would require three instructions: read the port to a CPU register, add the constant to the CPU register, and write the result back to the port.

I strongly recommend that you read that wiki article for further information.

To answer one of your questions:

What or where am I writing to if it's not in memory?

You are writing to the registers of the I/O interface through the data bus, which later (when ready) sends the data to the actual I/O device. Below is an image of an example I/O device interface.