Click into the customers file that you wish to keep - this will be the master record. Click on the [Cog] icon on the customers menu.

Once on the merge screen search for the duplicate customer. 

Select the customer you are going to merge into the master record.

On the next step you can choose to copy over the customers bookings and associated records. You can also copy over the duplicate pets to the master record.
Finally click on the [Merge Customers] button.

And the merge is complete! Good Job :)