"Alternate" again comes from the theory - it is any candidate key apart from the primary key. It is similar to SQL's Unique constraint, except that it is a true key (i.e. cannot contain nulls). So yes, you can simulate an alternate key in Oracle using a combination of NOT NULL and UNIQUE constraints.
A table can easily have 2 or more candidate keys: for example, consider this table for hotel room reservations:
CREATE TABLE reservation
( reservation_no INTEGER NOT NULL
, room_no INTEGER NOT NULL
, from_date DATE NOT NULL
, to_date DATE NOT NULL
There are 3 candidate keys here - i.e. three sets of columns whose values must always be unique to a single row:
2) room_no, from_date
3) room_no, to_date
If we choose reservation_no to be the primary key, then the other 2 are alternate keys. (But we could choose any of the 3 as the primary).
In Oracle, we would define these 3 keys like this:
ALTER TABLE reservation ADD
( CONSTRAINT res_pk PRIMARY KEY (reservation_no)
, CONSTRAINT res_uk1 UNIQUE (room_no, from_date)
, CONSTRAINT res_uk2 UNIQUE (room_no, to_date)