How to unlock an Xperia z1 when password has been forgotten?

You have to hard reset your Xperia Z1. NOTE: Backup your files because a hard reset will erase ALL your data and settings (copy them to your PC or somewhere else). You will have to set up your phone from the beginning.

1. Open the SIM card slot.
2. There you'll se a little red button. Press it with a toothpick or something thin and hold it for 3-5 seconds.

If this doesn't work, try this:
NOTE: Before doing this, your battery must be half or more full (as in previous method).
1. Turn off your phone.
2. Reboot phone by pressing the power button and volume up botton together.
3. Using volume keys, navigate to Wipe Data/Factory Reset and confirm it with the power button.
4. Wait for the process to finish. You'll have to set the phone up from the beginning.

Actually I wouldn't do this, you will loose all your data especially if time do not perform a back up. Download and install Sony Pass lock Remover onto your PC. Follow the prompts it is very easy!