How does a transistor amplify a signal?
A transistor is a constructed from materials that behave as semiconductors--which is to say that these materials conduct electrical current but only when certain conditions are met. By combining these materials into a device called a transistor we are able to create an electronic component that will only conduct electricity when a small electric current is applied to the so-called Base pin of the transistor.

One obvious use of such behavior is to use the transistor as a switch--by applying or removing current to the base pin we are able to control whether the transistor conducts electricity via its collector and emitter pins. When no current is present at the Base pin the transistor will block electrical transmission through its emitter and collector pins.

Generally speaking, the electrical current required to control the transistor at the base pin is very small. Much larger currents can be transferred through the device through the collector and emitter pins. Thus, in effect, a transistor uses a very small input current to control the flow of a much larger current.

An audio signal, for example, is transmitted as a very low power electrical current--in it's original form it is too small to be able to drive a loudspeaker to make sounds that are audible to the human ear. To amplify this signal so that is usable for audio playback we can apply the audio signal to a transistor's base pin--this will allow a much larger current to flow between the emitter and collector pins and through the transistor. Since the flow of this large current will mirror the much smaller current of the audio signal we will have effectively created a much stronger version of the original input signal.