Generally there are (or could be many) USB Devices and (only one controlling) USB Host. The differences are discussed in this stackexchange.com question. Briefly, USB Hosts are usually very complex hardware and software devices that may know about many many different types of peripherals. They are usually only found on computers. USB Devices are usually dedicated peripherals. They only need to know about them selves. Many people do not realize that there are many protocols built into USB. These are normally called Device Classes. The USB Serial Protocol (or CDC) is one of dozens of Device Classes.
So, assuming you want a build a Serial Port USD Device, you would first need a processor with built in USB Device hardware. (There are likely some project out in the wild that have implemented this feature using software. It might be a good experience. But for practical purposes it is usually better to use hardware for this.) The next feature you need to implement is a software USB Device driver. This is not trivial. But is much simpler than the USB Host software. The next feature you need to implement the serial protocol inside the USB protocol.
In the real world, the USB code, even for a USB Device, is complex enough that most developers use the USB code provided by the processor manufacturer. This is very common as most manufacturers want you to use their hardware and provide the USB code for free. Here is Atmel's support page for CDC USB (serial port) support. (Of course, Atmel makes the processor found in most Arduino boards.)
Thus far this answer has been general and is likely difficult to implement in a practical sense. A more turn key solution is to use an existing platform which contains a USB CDC Device. The Arduino Uno comes in several versions. One has a USB CDC chip made by FTDI. Another has a 2nd processor programmed specifically to handle USB CDC type communications. Either version provides a USB CDC interface used for programming. I believe this web page explains how you can also use this port to communicate with your sketch.