Create a PostgreSQL constraint to prevent unique combination rows

You can use an exclusion constraint with btree_gist,

-- This is needed
CREATE EXTENSION btree_gist;

Then we add a constraint that says:

"We can't have 2 rows that have the same name and different is_active":

ALTER TABLE table_name
  ADD CONSTRAINT only_one_is_active_value_per_name
    EXCLUDE  USING gist
    ( name WITH =, 
      is_active WITH <>      -- if boolean, use instead:
                             -- (is_active::int) WITH <>
    );

Some notes:

• is_active can be integer or boolean, makes no difference for the exclusion constraint. (actually it does, if the column is boolean you need to use (is_active::int) WITH <>.)
• Rows where name or is_active is null will be ignored by the constraint and thus allowed.
• The constraint makes sense only if the table has more columns. Otherwise, if the table has only these 2 columns, a UNIQUE constraint on (name) alone would be easier and more appropriate. I don't see any reason for storing multiple identical rows.
• The design violates 2NF. While the exclusion constraint will save us from update anomalies, it may not from performance issues. If you have for example 1000 rows with name = 'A' and you want to to update is_active status from 0 to 3, all 1000 will have to be updated. You should examine whether normalizing the design would be more efficient. (Normalizing meaning in this case to remove is_active status from the table and add a 2-column table with name, is_active and a unique constraint on (name). If is_active is boolean, it could be totally stripped and the extra table just a single column table, storing only the "active" names.)

This is not a case where you can use a unique index. You can test the condition in a trigger, e.g.:

create or replace function a_table_trigger()
returns trigger language plpgsql as $$
declare
    active int;
begin
    select is_active into active
    from a_table
    where name = new.name;

    if found and active is distinct from new.is_active then
        raise exception 'The value of is_active for "%" should be %', new.name, active;
    end if;
    return new;
end $$;

Test it here.