A series circuit has more than one element (anything that uses electricity to do work) and gets its name from having only one path for the charges to move along. Charges must move in series first going to one element, then the next. If one of the elements in the circuit is broken, then no charge will move through the circuit because there is only one path. There is no alternative route.
Consider three bulbs connected one after the other. If you look closely inside of the bulb, you will notice that it has a tungsten wire coiled on two metal connectors. This tungsten coil provides the resistance when current is passed through the bulb. Due to the resistance, the tungsten coil heats up and emits light. This is an example of a simple series circuit. The 6 V battery provides 2 V across each lamp. Three bulbs are connected in series with a battery. The same current exists almost immediately in all three lamps when the switch is closed. The charge does not "pile up" in any lamp but flows through each lamp.
Electrons in all parts of the circuit begin to move at once. Some electrons move away from the negative terminal of the battery, some move toward the positive terminal, some move through the filament of each lamp. Eventually the electrons move all the way around the circuit (the same amount of current passes through the battery). This is the only path of the electrons through the circuit. A break anywhere in the path results in an open circuit, and the flow of electrons ceases. Burning out one of the lamp filaments or simply opening the switch could cause such a break.