By default a regular user does not have permission to use the serial port. In my case I wanted to use two USB-to-serial adapters:
$ ls -la | grep ttyUSB
crw-rw---- 1 root dialout 188, 0 May 11 11:02 ttyUSB0
crw-rw---- 1 root dialout 188, 1 May 11 11:02 ttyUSB1
By looking at output you can see that the owning group is dialout. So it is a matter of adding your user to that group (and logout/login as with every user rights change). The bash environment variable $USER contains your username.
$ sudo adduser $USER dialout
Found on the Ubuntu help pages.