Economies of scale are where you can buy multiple items for a lower per-unit price than buying single items.

Economies of scale exist in part due to the interaction of Fixed costs and Variable costs. (Note: I take fixed costs as 'costs that are fixed for a production run of any size' and variable costs as 'costs that vary with production run size'. Some people take fixed costs as 'fixed unit cost, regardless of production run size' and variale costs as 'fraction of machine costs per unit'. If you like, you can switch the words round.)

An example: Company X wants to make injection-moulded plastic widgets. They face two types of cost, fixed and variable. The fixed costs don't change much as the number of units produced increases, i.e. the cost of the injection moulding machine, rent on the factory, etc. Let's say the machine costs £10,000. The fixed costs will be the same for one unit or one thousand.

They also face Variable costs. These are the costs that vary depending on how many things they produce, i.e. the cost of plastic. Let's say each widget takes £5 or plastic.

The variable cost for each item is: £5
The fixed cost for each item is (Fixed costs)/(Number of items produced)

So, the cost of one unit is the variable costs, plus a part of the fixed costs. Consider this table:

Widgets    |   Variable   |   Fixed     |  Total 
produced:  |   cost per   |   cost per  |  cost per
           |   widget:    |   widget:   |  widget:
-----------+--------------+-------------+-----------
1          |   $5         |   £10,000   |  £10,005
10         |   £5         |   £1,000    |  £1,005
50         |   £5         |   £200      |  £205
100        |   £5         |   £100      |  £105
1,000      |   £5         |   £10       |  £15
2,000      |   £5         |   £5        |  £10
10,000     |   £5         |   £1        |  £6
           |              |             |

In this situation, it is cheaper per widget to produce 10,000 widgets, since it divides up the cost of the machine. So if you went to this company with your widget design and asked them to make you one, it would cost a lot per widget, but if you asked them to make 10,000 it would cost much less per widget.

Here is a graph:

   | *
   | *
   | **
 C |  *
   |   *
   | !  **
 O | !    **
   | !!     ***
   |  !        ***
 S |   !          *****
   |    !!            **********************
   |++++++!!++++++++++++++++++++++++++++++++
 T |        !!!
   |           !!! 
   |             !!!!!
   |                  !!!!!!!!!!!!!!!!!!!!!!
   +-----------------------------------------
      P R O D U C T I O N   Q U A N T I T Y

! - Variable Costs
+ - Fixed Costs
* - Total Cost

The situation described above is an example of Manufacturing Economies of Scale. There are other types, as enumerated by Noung above.

The opposite would be diseconomies of scale.