Oracle Equivalent to MySQL INSERT IGNORE?

Check out the MERGE statement. This should do what you want - it's the WHEN NOT MATCHED clause that will do this.

Do to Oracle's lack of support for a true VALUES() clause the syntax for a single record with fixed values is pretty clumsy though:

MERGE INTO your_table yt
USING (
   SELECT 42 as the_pk_value, 
          'some_value' as some_column
   FROM dual
) t on (yt.pk = t.the_pke_value) 
WHEN NOT MATCHED THEN 
   INSERT (pk, the_column)
   VALUES (t.the_pk_value, t.some_column);

A different approach (if you are e.g. doing bulk loading from a different table) is to use the "Error logging" facility of Oracle. The statement would look like this:

 INSERT INTO your_table (col1, col2, col3)
 SELECT c1, c2, c3
 FROM staging_table
 LOG ERRORS INTO errlog ('some comment') REJECT LIMIT UNLIMITED;

Afterwards all rows that would have thrown an error are available in the table errlog. You need to create that errlog table (or whatever name you choose) manually before running the insert using DBMS_ERRLOG.CREATE_ERROR_LOG.

See the manual for details

If you're on 11g you can use the hint IGNORE_ROW_ON_DUPKEY_INDEX:

SQL> create table my_table(a number, constraint my_table_pk primary key (a));

Table created.

SQL> insert /*+ ignore_row_on_dupkey_index(my_table, my_table_pk) */
  2  into my_table
  3  select 1 from dual
  4  union all
  5  select 1 from dual;

1 row created.