Generically to transfer data, say 8bits:
parallel requires 8 wires, one for each bit of the data so that all 8bits are sent at the same time.
serial requires only one wire and each of the 8bits is send one at a time.
If you are speaking of specific interfaces such as on a PC then the generic definition holds but there are additional definitions.
So, on a PC the parallel interface is usually called the" Printer Port" and was originally for a Centronics Printer (google it for the details).
A serial interface of a PC was originally RS232 (again google it for details) for connecting MODEMs.
Both of these interfaces has evolved but should be back-ward comparable with older equipment.
I *think* serial connects directly to UART?
It may or may not. To answer this I need to know exactly what "serial" and what UART you need to connect. Look up the RS232 details and you'll see what the voltage levels are and what the logic levels are for one's and zero's. Then look at the data sheet for the UART and see what its voltage and logic levels are.
If you wish to connect the UART from a uProcessor to the COM port on a PC then you need a level translator circuit between them. The common IC to use is the MAX232 or MAX3232. There are many posts here on SoR on serial interfacing a uprocessor and PC. Also look at the SoR tutorials for how to do this.
Does this answer your questions?