Basic operation of a bipolar junction transistor
When electrons flow through a forward-biased diode junction, such as the base-emitter junction of a transistor, it actually takes a non-zero amount of time for them to recombine with holes on the P side and be neutralized.
In an NPN transistor, the P-type base region is constructed so as to be so narrow that most of the electrons actually pass all of the way through it before this recombination occurs. Once they reach the depletion region of the reverse-biased base-collector junction, which has a strong electrical field across it, they are quickly swept away from the base region altogether, creating the collector current.
The total current through the base-emitter junction is controlled by the base-emitter voltage, which is independent of the collector voltage. This is described by the famous Ebers-Moll equation. If the collector is open-circuit, all of this current flows out the base connection. But as long as there's at least a small positive bias on the collector-base junction, most of the current is diverted to the collector and only a small fraction remains to flow out of the base.
In a high-gain transistor, fewer than 1% of the electrons actually recombine in the base region, where they remain as the base-emitter current, which means that the collector current can be 100× or more the base current. This process is optimized through careful control of both the geometry of the three regions and the specific doping levels used in each of them.
As long as the transistor is biased in this mode of operation, a tiny change in base-emitter voltage (and a correspondingly small change in base-emitter current) causes a much larger change in collector-emitter current. Depending on the external impedance connected to the collector, this can also cause a large change in collector voltage. The overall circuit exhibits power gain because the output power (ΔVC × ΔIC) is much greater than the input power (ΔVB × ΔIB). Depending on the specific circuit configuration, this power gain can be realized as either voltage gain, current gain, or a combination of both.
Essentially the same thing happens in a PNP transistor, but now you have to think of the holes (the absence of an electron) as being the carrier of a positive charge that drifts all of the way through the N-type base to the collector.
Read and re-read Dave's excellent answer.
Then mentally reverse what's going on...
You have a forward-biased base-emitter junction, and external circuitry connected to the base demands a current Ib, which is supplied from electrons sourced by the emitter.
But when an electron enters the base region, it encounters a strong electric field pulling it towards the (positive) collector. The majority (a large and quite well defined proportion) of these electrons are lost (from the base current) and emerge as collector current, for the reasons explained so well in Dave's answer. So rather than an efficient amplifier, you could equally well view the transistor as a hopelessly inefficient supplier of base current!
From this viewpoint, the base circuit demands Ib and the emitter supplies it. But as a byproduct, a much larger current (Ic = 100Ib) is "lost" to the collector. Which is of course what we really want.
EDIT re: comment: Ultimately (most of, say 99%) the electrons from emitter enter the collector region.
Ultimately the collector current has to be (slightly) smaller than the supply emitter current.
Right to both of these.
What is the purpose?
1) A very small base current controls a large collector current, and the emitter current is the sum of these two.
2) The ratio Ic/Ib (hFE or current gain) is approximately independent of the collector voltage Vce (until Vce is low, say < 1V). This means that for a suitable choice of impedance in the collector circuit, a small change in Ib can result in a large change in Ic and a large change in Vce; this is where voltage gain comes from.
So the usual "common emitter" amplifier has the load in the collector circuit and has both high current gain and high voltage gain.