When there are several components in an electric circuit, the voltages and currents in different parts of the circuit depend on how the components are connected. They can be connected in series, parallel or a series/parallel combination. Series components are connected end-to-end, forming a single path for the current. A component connected in parallel has its ends connected to the ends of all the other parallel components so that there are multiple paths. Most real-life circuits are a combination of series and parallel branches.
The resistances of components connected in series add up to create a combined resistance equal to the sum of the individual resistances. For components with resistances R1, R2, R3, etc., the total resistance is R = R1 + R2 + R3, etc. For a voltage "E" across the components, the current "I" through the components will be I = E/R. The voltage across R1 will be E1 = I x R1; E2 across R2 will be I x R2; and so on. To illustrate, for E=12 volts, R1 = 10 ohms and R2 = 20 ohms, R = 10 + 20 = 30 ohms. The current I = 12/30 = 0.4 amps. E1 = 0.4 x 10 = 4 volts and E2 = 0.4 x 20 = 8 volts. The higher resistance sees the proportionally higher voltage.
Components connected in parallel have a lower total resistance than the individual components. For resistances R1, R2, R3, the total resistance 1/R = 1/R1 + 1/R2 + 1/R3. For a voltage "E" applied to the parallel components, R1 will have a current of I1 = E/R1; R2 will have a current of I2 = E/R2; and R3 will have a current of I3 = E/R3. The total current in the circuit will be I = I1 + I2 + I3. As an example, using the same resistors and voltage as in Section 2, 1/R=1/10+1/20. Then 1/R=3/20 and R = 6.67 ohms. I1 = 12/10 = 1.2 amps and I2 = 12/20 = 0.6 amps. Both resistances see the same voltage but the higher resistance sees a proportionately lower current.
Real-life circuits are complex in the sense that they have many components, some connected in series and some in parallel. Every such circuit breaks down into circuit branches that are connected in series and other branches that are connected in parallel. Adding the resistances in each branch as described in Sections 2 and 3 allows you to replace each branch with a single equivalent resistance. The circuit simplified in this way allows you to calculate the remaining series and parallel resistances and the currents and voltages across each component in the circuit, using the methods of Sections 2 and 3.