Series Circuits
In a series circuit, each component of the circuit (battery, resistors, etc.) is connected by wires from one to the next in a line. They all make one big loop without any branches or other complications. The electrons traveling through the circuit begin at the battery and move through every wire and component. If any of the components fail and cause a break in the circuit, such as a Christmas tree light burning out, electricity cannot flow anywhere in the whole circuit. All the lights in that string go out until you find and replace the faulty bulb.
Parallel Circuits
In a parallel circuit, the battery is connected directly to each component. For example, wires may go from each terminal of a battery and branch into three to connect to either end of three light bulbs. This creates three alternate paths that any given electron might take on its journey from one battery terminal to the other. If one of the light bulbs goes out, electricity stops flowing in that branch, but it can still flow through the other two. In a home, this means that when one light burns out they don't all stop working.
Current
In a series circuit, the current through the circuit is the same the whole way around. Since every electron must follow the same path, the current cannot change anywhere in the circuit. In a parallel circuit, however, the current is divided when the circuit branches. The current in one branch is different than the current in another branch. In the places where the branches converge, the current is the sum of the current through each branch.
Voltage
In a series circuit, the voltage changes every time the electrons encounter a different component. Since each component has a different resistance, the voltage between the wire before the component and the wire after it must be different. In a parallel circuit, however, the voltage is the same for every parallel component. All of them are effectively connected directly to the battery terminals, so they must have the same voltage between their two ends.