MYSQL speed up query

First, as others have said if you have the possibility you should really revise the structure of your database, what you have now doesn't make much sense.

However I've been in that "having to do reports from existing code" situation so I know that you might not be able to do that at all; if that is indeed your case then the following query should help you significantly improve speed:

SELECT 
  `Record_Number`, 
  Concat(Name, ' ', Group_Concat(Value ORDER BY rn ASC SEPARATOR ' ')) AS Request
FROM 
  `Ordered_Items` oi 
  LEFT OUTER JOIN `Ordered_Options` oo ON (oo.Ordered_Items_rn = oi.Record_Number) 
GROUP BY 
  oi.Record_Number

Which should give you something like:

Record_Number   Request
1               Pizza Jane Doe Pepperoni
2               Fries Bobby Table French

Since you are new around here, please remember to accept as a solution by cliking on the arrow on the left is this fixes your problem. If it doesn't, please indicate why in a comment to this answer.

Sample database I used for it:

--
-- Table structure for table `Ordered_Items`
--    
CREATE TABLE IF NOT EXISTS `Ordered_Items` (
  `Record_Number` int(11) NOT NULL,
  `Name` text NOT NULL
) ENGINE=MyISAM DEFAULT CHARSET=utf8;

--
-- Dumping data for table `Ordered_Items`
--

INSERT INTO `Ordered_Items` (`Record_Number`, `Name`) VALUES
(1, 'Pizza'),
(2, 'Fries');

-- --------------------------------------------------------

--
-- Table structure for table `Ordered_Options`
--

CREATE TABLE IF NOT EXISTS `Ordered_Options` (
  `Ordered_Items_rn` int(11) NOT NULL,
  `Value` text NOT NULL,
  `rn` int(11) NOT NULL
) ENGINE=MyISAM DEFAULT CHARSET=utf8;

--
-- Dumping data for table `Ordered_Options`
--

INSERT INTO `Ordered_Options` (`Ordered_Items_rn`, `Value`, `rn`) VALUES
(1, 'Jane', 43),
(1, 'Doe', 44),
(1, 'Pepperoni', 45),
(2, 'Bobby', 43),
(2, 'Table', 44),
(2, 'French', 45);

A few comments:

• Normalize your table, no amount of tricks will speed up this unnormalized mess to acceptable proportions.
• Get out of the ` backtick habit, it's an eyesore. You only need to backtick reserved words or column names with -gasp- spaces.
• You don't need to use () parentheses to this extend, as long as only use AND and no subselects you don't need any.
• Indenting is good, indenting using tab is bad, especially when pasting code on stackoverflow.

With regards to your table.

Throw out the one you have now and create a new table.

table ordered_item
-----------
id integer auto_increment primary key  
item_id integer not null foreign key references item(id) on delete restrict on update cascade
customer_id integer not null foreign key references customer(id) on delete restrict on update cascade
qty integer not null
.....

table customer
--------------
id integer auto_increment primary key    
firstname varchar(200)
.....

table item
----------
id integer auto_increment primary key  
name varchar(200)
price decimal(10,2)  //should really be in a separate table

The following query will give all items ordered per order with a grand total for customer 1124.

SELECT customer.name, io.id as order_id, sum(io.qty) as number_of_items, i.* 
FROM item i
INNER JOIN ordered_item oi ON (oi.item_id = i.id)
INNER JOIN customer c ON (c.id = oi.customer_id)
WHERE customer.id = '1124'
GROUP BY io.id WITH ROLLUP

This is the way SQL is supposed to work, not the way you are doing it.

BTW the brackets are optional, I just like 'm for readability.