I doubt that a password would be a key. Most systems that I know of don't require passwords to be unique.

In SQL Server a unique index on non-nullable columns is just a key by another name. Unique indexes and uniqueness constraints are simply two different ways to create keys. I would not recommend using a unique index instead of a uniqueness constraint. Constraints, not indexes, are the conventional, SQL standard and most obvious way to implement keys. As far as I know unique indexes alone have no particular advantages over uniqueness constraints. Was there a reason why you mentioned unique indexes specifically? What do you mean by a "non-keyed" index?