Syntax error in simple trigger

I assume you will value TableB first as it contains the order_no. In that case you need to use an update statement in your trigger instead of an insert statement:

SQL Fiddle

MySQL 5.6.6 m9 Schema Setup:

CREATE TABLE TableA(order_id INT, sku VARCHAR(10));

CREATE TABLE TableB(order_id INT, order_no VARCHAR(10),sku_copy VARCHAR(10));

GO
CREATE TRIGGER trigger_name
    AFTER INSERT ON TableA
    FOR EACH ROW BEGIN

    UPDATE TableB
    SET sku_copy = NEW.sku
    WHERE order_id = NEW.order_id;
END;
GO
INSERT INTO TableB(order_id, order_no)VALUES(1,'111');
INSERT INTO TableB(order_id, order_no)VALUES(2,'222');
INSERT INTO TableB(order_id, order_no)VALUES(3,'333');
GO
INSERT INTO TableA(order_id, sku)VALUES(1,'AAA'),(2,'BBB');

(The GO in this example is used as a batch separator and not send to MySQL.)

Query 1:

SELECT * FROM TableB;

Results:

| ORDER_ID | ORDER_NO | SKU_COPY |
----------------------------------
|        1 |      111 |      AAA |
|        2 |      222 |      BBB |
|        3 |      333 |   (null) |

If you also want to react to updates, just create an AFTER UPDATE trigger like this:

SQL Fiddle

MySQL 5.6.6 m9 Schema Setup:

CREATE TABLE TableA(order_id INT, sku VARCHAR(10));

CREATE TABLE TableB(order_id INT, order_no VARCHAR(10),sku_copy VARCHAR(10));

GO
CREATE TRIGGER TableA_AfterInsert
    AFTER INSERT ON TableA
    FOR EACH ROW BEGIN

    UPDATE TableB
    SET sku_copy = NEW.sku
    WHERE order_id = NEW.order_id;
END;
GO
INSERT INTO TableB(order_id, order_no)VALUES(1,'111');
INSERT INTO TableB(order_id, order_no)VALUES(2,'222');
INSERT INTO TableB(order_id, order_no)VALUES(3,'333');
GO
INSERT INTO TableA(order_id, sku)VALUES(1,'AAA'),(2,'BBB');
GO
-- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- 
GO
CREATE TRIGGER TableA_AfterUpdate
    AFTER UPDATE ON TableA
    FOR EACH ROW BEGIN

    IF (OLD.sku != NEW.sku)
    THEN
      UPDATE TableB
      SET sku_copy = NEW.sku
      WHERE order_id = NEW.order_id;
    END IF;
END;
GO
UPDATE TableA
  SET sku = 'NEW'
WHERE order_id = 2;
GO

Query 1:

SELECT * FROM TableB;

Results:

| ORDER_ID | ORDER_NO | SKU_COPY |
----------------------------------
|        1 |      111 |      AAA |
|        2 |      222 |      NEW |
|        3 |      333 |   (null) |

In both cases the NEW and OLD virtual tables refer to the table the trigger is defined on. NEW contains the new version of the row that was inserted or changed. OLD contains the pre-change version of the row. OLD is only defined in an update trigger as there is no old version on an insert.

^{(Obvious error in the trigger logic aside.)}
In Postgres 9.2 or later, use the function pg_trigger_depth() that @Akash already mentioned in a condition on the trigger itself (instead of the body of the trigger function), so that the trigger function is not even executed when called from another trigger (including itself - so also preventing loops).
This typically performs better and is simpler and cleaner:

CREATE TRIGGER set_history
BEFORE UPDATE ON field_data
FOR EACH ROW 
WHEN (pg_trigger_depth() < 1)
EXECUTE PROCEDURE gener_history();

The expression pg_trigger_depth() < 1 is evaluated before the trigger function is entered. So it evaluates to 0 in the first call. When called from another trigger, the value is higher and the trigger function is not executed.