What does a CONSTRAINT have to do with my unique index? I have to indexes on my table.  The first was created by Django and the second by me. I'm not completely certain what the extra CONSTRAINT means in the first index and am wondering how I would change my CREATE INDEX statement to add that to it.

"customers_phonetype_customer_id_176731583d230ba5_uniq" UNIQUE CONSTRAINT, btree (customer_id, display) ;-- Django created
"customers_phonetype_customer_id_uniq" UNIQUE, btree (customer_id, lower(display::text)) ; -- Manually created


This is how I manually created the second index:

create unique index customers_phonetype_customer_id_uniq on customers_phonetype (customer_id, lower(display));


What is the meaning of the extra word CONSTRAINT in this case? I'm not seeing it in the docs.
The first one is a unique constraint. It can be added to an existing table with:

ALTER TABLE ADD CONSTRAINT ...


Details in the manual here.
It is implemented using a unique index. Per documentation:


  Adding a unique constraint will automatically create a unique btree
  index on the column or group of columns used in the constraint. A
  uniqueness constraint on only some rows can be enforced by creating a partial index.


In addition it allows foreign key references to it.

The second one in a unique index.
It couldn't be a unique constraint because those only allow columns, not expressions. More details:


How does PostgreSQL enforce the UNIQUE constraint / what type of index does it use?
Discussion on pgsql-general about the difference between constraint and index.