Confusion on ARM-Tiva PORTS and Registers
A port is a group of pins -- typically eight or sixteen of them.

A register is a location in memory which controls a peripheral on the microcontroller. Some of those peripherals are GPIO peripherals, which allow you to control I/O on pins. Each port has its own GPIO peripheral (e.g, GPIOA, GPIOB, etc).

For instance, a typical GPIO peripheral will have one register to control whether each pin is an input or an output, one register to control what data is being output, one read-only register to represent the input data, and probably at least a couple other registers to control pull-up/down, slew speed, AFIO selection, and whatnot.

"Serial ports" and "parallel ports" are terms from personal computers. They don't really apply to microcontrollers; there are a variety of commonly available I/O peripherals which can be used in serial and/or parallel modes.

(What you're thinking of as a "serial port" is usually represented in a microcontroller as a UART or USART peripheral. They can be connected to pins which are "borrowed" from GPIO peripherals, but typically have quite a lot of different registers internally.)

The table in the image you've linked is… kind of useless. You should probably take a look at the datasheet for the microcontroller you're using; it'll probably explain things a lot better than that table.