Parallel and Series Circuits
Series Circuit with 3 lamps.
Parallel Circuit with 3 lamps.
Parallel Circuit with 2 lamps and 3 ammeters. Series Circuit with 2 lamps and 3 ammeters.
Parallel Circuits
In a parallel circuit, there are junctions placed so that there are multiple paths for the current to follow.  As the number of bulbs increase, so does the amount of current flowing through them, because the resistance is spread throughout the paths.  Also, the current is not the same at all points in the circuit.  The main advantage of a parallel circuit system is that if one item (such as a bulb) shorts out, the rest of the bulbs will stay on.

Series Circuits
In a series circuit, there are no wire junctions, and all bulbs are placed consecutively on the same path.  The more bulbs there are in a series circuit, the more resistance to flow tthere will be, meaning less overall current in the circuit.  Thus, you can regulate the amount of current not to overflow by simply adding more bulbs.  The advantage to a series circuit system is that if a necessary item shorts out, the whole system will shut down rather than run dangerously.

Calculating Resistance and Current
The main formula in calculating resistance and current is E=IR; where E stands for voltage, measured in volts; I stands for current, measured in amperes (amps); and R stands for resistance, measured in ohms.
In a series circuit, each bulb's resistance adds up to create a total resistance, and you then divide the voltage in the circuit by the total resistance to arrive at the current for the circuit.  For example, if a series circuit has a 9-volt battery and 3 1-ohm resistors connected, then the total resistance is 3 ohms, and the total current is 9 / 3 = 3 amps.
In a parallel circuit, each bulb's current adds up to a create a total current, and you then divide the voltage in the circuit by the total current to arrive at the resistance of the circuit.  For example, if you have a 120-volt circuit with 3 10-amp bulbs, then the total current is 30 amps, and the total resistance is 120 / 30 = 4 ohms.

by Pradeep

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Sources:
Parallel and Series Circuits @ http://www.jec.co.uk/Community___the%20_Environment/Education_Resources/Simple_Circuits/Parallel_and_Series_Circuits/parallel_and_series_circuits.html
Map Circuits @ http://mit.concord.org/4th/sc4map-circuits.html
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