The schematic symbols for pnp- and npn-type BJTs.
Conceptually, one can understand a bipolar transistor as two diodes placed back to back. In normal operation, the emitter-base junction is forward biased and the base-collector junction is reverse biased. In an npn-type transistor for example, electrons from the emitter wander (or "diffuse") into the base. These electrons in the base are in the minority - there are plenty of holes to recombine with. The base is always made very thin such that most of the electrons diffuse over to the collector before they recombine with holes. The collector-base junction is reverse biased to prevent the flow of holes, but electrons meet a more friendly reception - they are swept into the collector by the electric field around the junction. The proportion of electrons able to run the base "gauntlet" and make it to the collector is very sensitive to the voltage applied to the base. Hence, a small change of the base-emitter voltage can translate to a large change in electron flow between emitter and collector.
Cross section of an npn-type BJT as found in integrated circuits (simplified). The base region is the pink layer sandwiched between the green emitter region above and the yellow collector volume below.
Transistors have different regions of operation. In the "linear" regime, collector-emitter current is approximately proportional to the base current but many times larger, making this the ideal mode of operation for current amplification. The BJT enters "saturation" when the base current is increased to a point where the external circuitry prevents the collector current from growing any larger. A residual voltage drop of approximately 0.3V between collector and emitter then remains. A transistor is said to operate in the "cut off" regime when the base-emitter voltage is too small for any significant current to flow. In typical BJTs manufactured from silicon, this is the case below 0.7V or so. BJTs that alternate between cut off and saturation can by viewed as electronic switches.
See also electronics