Which field type should I use for a telephone number, and how can I ensure each entry is unique? I have created a field (field_telephone) to store telephone numbers like 13712345678.

Which is the best field type to use?
I also want to make sure the values for this field are unique. How can I do that?
For a basic telephone field without validation you could just use a text field, and set the maximum length to something sensible for telephone numbers in the regions you're targeting.

If you want a more advanced telephone field, check out the Phone module. There's a stable release for Drupal 6, but only a dev release for Drupal 7.

To make sure fields are unique, you can use the Unique Field module.