Equivalent resistance is a handy thing to know in a few situations. One situation is when your college professor is testing your knowledge on equivalent resistance and you want to pass. In this instance, you will have to reduce a confusing circuit filled with lots of resistors down to a circuit with fewer resistors. And then you solve lots of equations on this smaller circuit. However, in real life, this rarely happens.
What will be more likely, however, is to connect two resistors in series or parallel to get a desired resistance. For example, for a hobby project I was working on, I needed a 130 ohm resistor and I couldn't find any at Active Electronics, my local electronics shop. But they had plenty of 260 ohm resistors. Putting two of them in parallel got me 130 ohms and my circuit worked great.
It might seem silly, but resistors don't come in every value you need, despite how many letters you write to the manufacture. If you can't find the exact resistance you need, you have a few options: redesign your circuit to fit the resistors you can get a hold of, special order resistors to the exact values you need, which could be costly, or find the equivalent resistance of some resistors.
Below is an explaination of series and parallel resistance.
Series Resistance
For resistors to be in series with each other, the end of one resistor is connected to the beginning of another resistor and only that resistor, as shown below.
These two resistors are in series.
/\/\/\/\/\/\
R1 R2
When resistors are in series, a common current is passed through them. Meaning the current across each resistors in series is the same. The voltage across each resistor, however, are different. To calculate the voltage, use the equation V = I * R.
Applying Kirchhoff's Voltage Law to the two resistors gives the following equation: V = I*R1 + I*R2. Since the currents are the same, we can tidy up the equation by moving the I to the outisde to get: V = I * (R1 + R2). The voltage of the circuit depends on the current multiplied by the sum of the resistors. The resistors can then be represented with an equivalent resistor, Req, which is equal to the sum of the resistors. And the circuit can be redrawn replacing the resistors with the equivalent resistance as shown below:
Here is the equivalent circuit
/\/\/\
Req
This does not change how a circuit fuctions, but it helps reduce the number of resistors and for any hobbiest who doesn't want to special order resistors, it is an inexpense and easy way to get that resistance you need.
The equivalent resistance can be applied to any number of resistors as long as all the resistors are connected in series. The sum of all the resistors in series with each other can be represented as a single equivalent resistance.
Parallel Resistance
Resistors are connected in parallel when a common voltage is applied across each element. This means the top of each resistor in parallel are connected together and the bottom of each resistor in parallel are connected together.
These two resistors are connected in parallel

 
\ \
/ R1 / R2
\ \
 

This time the resistors are connected in parallel and thus they all share the same voltage, but each has a different current. Applying Kirchhoff's Current Law to the two resistors gives the following equation: I = V / R1 + V / R2. Since the voltages are the same, we can simplify the equation by moving V to the outside to get: I = V( 1/R1 + 1/R2). The current across the circuit depends on the voltage multiplied by the sum of the reciprocals of each resistor.
The resistors can now be replaced with a single equivalent resistor, Req, which is equal to one over the sum of the reciprocal of each resistors, Req = 1/( 1/R1 + 1/R2 ). Or the product of all the resistors divided by the sum of all the resistors. In our example there were only two resistors in parallel so the equivalent resistance is R1 * R2 /(R1+R2)
The circuit can be redrawn by replacing the resistors with the equivalent resistance as shown below:
Here is the equivalent circuit


\
/ Req
\


A quick rule of thumb is, resistors in series add their resistance together, and resistors in parallel reduce their equivalent resistance.
It is interesting to note, the more resistors connected in series, the higher the equivalent resistance becomes. But the more resistors connected in parallel, the lower the equivalent resistance becomes.