I have one custom made device for measuring current. This device can be connected to PC and communication is done over USB. It is based on FTDI chip. I have application writ
I think there's currently only one properly maintained library for serial USB communication and it's felHR85/UsbSerial. It supports several USB chips including FTDI (see Devices Supported).
Unfortunately mik3y/usb-serial-for-android seems to be dead.
And no, you don't need to root your device for connecting to and communicating with a USB device.
EDIT: We have a maintained fork of mik3y's library. See kai-morich/usb-serial-for-android
EDIT2: It seems that all of these libraries are maintained again, mik3y joined forces with kai-morich and they are working together.
There is a project dedicated to serial communication on Android. android-serialport-api. I think, it is a good resource to start with.
FTDI also provides Android related resources.
I can really recommend the USB-Serial-For-Android library, it supports multiple popular usb-to-serial chips (also FTDI) and there is no rooting required. Check out the GitHub project here.