In order to use the Edison's UART you have to use the pins 0 and 1 (Rx and Tx) just like in the picture you posted. The Rx pin is the reception line and the Tx pin is the transmission line. The device that you want to interface should have a UART interface as well. You should connect the Tx line of the Edison to the Rx line of the device, and the Rx line of the Edison to the Tx line of the device. Make sure that the other device you want to interface is 5V compatible.
Tx ------------------------> Rx
Rx <----------------------- Tx
Edison doesn’t use the apt command, that's why you got a "command not found" message. In order to install packages you have to use the opkg command, for example opkg install minicom. If you are not able to install the package you need, then you should look for a repo where the package is and install it from there. A good opkg repo made for Edison is the one shared in this thread: Edison opkg package repo created. I'm not sure if the repo includes the minicom package but it does include the screen package which is a similar package as minicom.
Thank you for you answer,I solved this problem. I want to ask you another question: when I use sudo apt-get install minicom,there is always an error,command not found. Can you tell me how to install minicom on Edison?
I would suggest that you install "screen" from the opkg repo. Screen is really easy to use as a serial terminal, here is an example :
screen /dev/ttyUSB0 19200