Electrolytes are dissolved ions. These allow the flow of electricity between a positive anode and a negative cathode in a wet cell battery.  Aqueous solutions of ionic compounds such as NaCl or CaCl2 are able to conduct electricity. 
Not all chemical compounds behave this way, however. If sugar is dissolved in water the solution that results is not electrically conductive.  It is not really difficult to imagine why this happens.  In order for a solution to conduct electricity there must be electrical charges in the water that are able to move.  When an ionic compound is dissolved in water, the ions that are packed together in the solid become separated . This is called dissociation.   As the ions enter the solution they become surrounded by water molecules that are attached to them.  This process is called hydration and we say that the ions become hydrated.  These hydrated ions are then able to migrate about in the solution and behave more or less independantly, and it is this ability to move that accounts for the electrical conductivity of the solution.   When a molecular substance like sugar which is made from  covalent bonds, the molecules disperse through the water but the molecules stay intact.  There are no charged particles in the solution, so the solution is a nonconductor.
The ability to produce a conducting solution is a special property of a solute -- a property that is given a special name.  Any solute that dissolves in water to give a solution that contains ions, and which therefore conducts electricity, is called an electrolyte.  On the other hand, a solute that remains undissociated in solution is called a nonelectrolyte.

Dissociation Reactions
When a solute dissolves, the ionic bonds holding it together break down and the solid becomes individual ions.  For example:

                      NaCl(s) -------->  Na+(aq)  +  Cl-(aq)

The aq, which stands for aqueous, means that the ions are dissolved in water and are hydrated, and by writing the formulas of the ions separately we mean that in the solution the ions are essentially independant of each other.  A similar equation that represents what occurs when calcium chloride, CaCl2(s), dissolves in water is:

                CaCl2(s) ------->  Ca2+(aq)  +  2 Cl-(aq)

which shows that two Cl- ions are present in the solution for each Ca2+. Once again we have used s and aq in parentheses to point out the changes that take place.
Salts that contain polyatomic ions also dissociate in aqueous solutions, and it is important to remember that in the solution the polyatomic ions remain intact. For instance, when the salt sodium sulphate, Na2SO4, dissolves in water, the solution contains sodium ions and sulphate ions.

                   Na2SO4(s)  -------->  2 Na+(aq)  +  SO42-(aq)

You can see that it is especially important that you know both the formulas and charges of the polyatomic ions to write equations such as these properly.
Often, in writing chemical equations for dissociation reactions or for other reactions involving ions in solution, the symbols (s) and (aq) following the formulas are omitted.  When we're discussing reactions in an aqueous solution, they are "understood."  Therefore, you should not be fooled when you see an equation such as

          CaCl2    ---------->   Ca2+   +  2 Cl-1

Unless something is said to the contrary, it means

         CaCl2(s)   ---------->   Ca2+(aq)  +  2 Cl-1(aq)


Strong and Weak Electrolytes
Some compounds when dissolved in water dissolve completely.  These compounds are considered to be strong electrolytes.   They are strong because they dissolve 100% of the time.  Some examples are HCl, or NaOH.
Other compounds do not dissolve very well in water. Vinegar or acetic acid dissolves in water. But it does not dissociate completely.  Actually vinegar only dissociates roughly 1.8%.  Therefore only 1.8% of the total amount of vinegar put into the water actually breaks down into ionic form.

                  CH3COOH  --------->  CH3COO-  +    H+
before             100%                            0                 0
after                98.2%                         1.8%           1.8%

         Go to the Colligative Properties of Water and Solutions Worksheet
         Go to the Dissociation Equations of Electrolytes Worksheet