Series Circuit
A series circuit connects components one after another along a single path, so the same current flows through each and one break stops all of them.
Definition
A series circuit has everything connected in one single loop, one part after another, so current has only one path and passes through every part in turn, and adding more bulbs makes every bulb dimmer since they all share the same push from the battery. Components are connected end to end along a single path so the same current flows through each; total resistance is the sum of the individual resistances, $R_{\text{total}} = R_1 + R_2 + \ldots$, and the source voltage divides among the components in proportion to their resistances, so if any component fails open, current stops in the entire circuit. Charge conservation forces equal current through every element, and the loop rule gives $\mathcal{E} = I\sum R_i$, so $R_{\text{eq}} = \sum R_i$; voltage across element $k$ is $V_k = \mathcal{E}\,R_k/R_{\text{eq}}$ (the voltage-divider rule) and its power is $I^2R_k$, so the largest resistance dissipates the most power, series capacitors combine as $1/C_{\text{eq}} = \sum 1/C_i$, the reverse of resistors, because they share charge rather than voltage, and internal resistance of a source or series-connected cells simply adds another element to the same sum.
Example
Old strings of holiday lights were wired in series, so one burned-out bulb broke the loop and darkened the whole string; three resistors of $2\ \Omega$, $3\ \Omega$, and $5\ \Omega$ in series with a $20\ \text{V}$ battery give a total resistance of $10\ \Omega$, so $I = 20/10 = 2\ \text{A}$ flows through each, with voltages of $4\ \text{V}$, $6\ \text{V}$, and $10\ \text{V}$ across them summing to $20\ \text{V}$. Four $1.5\ \text{V}$ cells, each with $0.2\ \Omega$ internal resistance, wired in series to a $4.2\ \Omega$ lamp give a total emf of $6.0\ \text{V}$ and total resistance of $5.0\ \Omega$, so $I = 1.2\ \text{A}$, the lamp receives about $6.0\ \text{W}$ and the cells waste about $1.2\ \text{W}$ as heat.
Key Insight
Series wiring is how a voltage divider works, splitting voltage in the ratio of the resistances, a trick used in nearly every sensor circuit. Series is the natural choice when every element must see identical current, such as LEDs in a strip driven by a constant-current supply, or the cells in a battery pack where a single weak cell throttles the entire string.