Many electrical components in an electric circuit have two leads (ends). As a result, they can be connected in one of two ways; in series (one electrical lead is touching the other), or in parallel (both leads are touching).
A series circuit provides exactly one path between any two points for electric current. These circuits have the advantage of making each component very dependent on the other components. This means that if one component is removed, all of the components turn off. While this is generally a bad idea (it would be obnoxious to have a light bulb turn off just because the TV was being turned off), there are a number of applications where this works better. Old style Christmas lights were like this, which is why if one bulb burned out, they all did. New strings of Christmas lights are slightly more complex which means that just the one bulb goes out, rather than the whole string.
Electric switches are in series with different electrical devices (in the case of a power strip, one switch can be in series with a number of electrical appliances in parallel) around the house. A light switch in series with an electric light will turn off the light upon opening. This will happen so quickly that humans can't even detect the time lag between flipping a switch and the light turning off.
Safety devices like circuit breakers and fuses are in series with the electric outlets in a house. All of the current that will flow through an electrical device must first flow through the circuit breaker (or fuse). If too much current is going to flow (potentially causing a fire), the circuit breaker breaks first (or the fuse blows). Since each circuit is in parallel, even if one circuit gets overloaded (tripping the circuit breaker or blowing the fuse), it won't have any effect on the other circuits. Likewise, different houses in a neighbourhood are parallel. A neighbour cooking dinner has no effect on someone ironing in a different house on a different circuit.
Different parts of the electrical grid tend to be in series. The transformer is in series with the distribution grid and the transmission system. This means that if any part of the electrical grid fails, people will experience a black-out. This has led to most electrical utilities (the people providing the electricity to the consumer) providing parallel backup paths for the electricity to get to the consumer.
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