From the official Ubuntu LostPassword documentation (with minor changes to work on more recent versions of Ubuntu)

• Reboot your computer.
• Hold Shift during boot to start GRUB menu.
• Press e to edit.
• Find the line starting with “linux” and change “ro” to  “rw” then append “init=/bin/bash” at the end of that line.
• Press Ctrl + X to boot.
• Type in passwd username.
• Type the new password
• Restart and login with the new password
• Enjoy!

Tested on Ubuntu 16.04 Desktop, 64-bit