BJT NPN current flow
It's not clear whether you're asking how a transistor works down at the physics level, or how it works in a circuit.  From the apparent level of your question, you should start out with the latter.  You don't actually need to know how a transistor does what it does on the inside to learn what it does from the outside world point of and to start using transistors in circuits.

Once you understand that, it is useful to have a basic grasp of the device physics, but there is a lot you can do with a simple circuit model.  Also, this is the electrical engineering site.  The internal workings of a transistor are better asked on the physics site.

Here is what a NPN BJT (bipolar junction transistor) does:



You can think of it as a adjustable valve that allows up to a certain amount of current to flow into the collector and out the emitter.  That valve is controlled by the base current, which also comes out the emitter.  The emitter current is therefore the base current plus the collector current.

So what's the big deal?  The main useful property is that the collector current the valve allows is proportional to the base current, but much larger than it.  The transistor has gain, which in this case is the ratio of collector current it will allow to the base current it takes to allow that collector current.

Small signal transistors easily have a gain of 50 or more.  Large power transistors maybe 15-30 typically.  There are some signal transistors that have guaranteed gains of several 100.

For example, let's say you have a transistor with a gain of 50.  If you ground the emitter and inject 1 mA into the base, then the collector can now sink up to 50 mA.

A couple more wrinkles, and you can actually design useful circuits with these things:

B-E the transistor looks like a diode to the circuit.  That means for a normal silicon transistor, the B-E voltage will be in the 500-700 mV range when you're putting useful control currents thru the base.  Just like a diode, the voltage is reasonably insensitive to the current, but there will always be some variation.

When the valve is adjusted to allow for more current than is being supplied into the collector, you would expect the C-E voltage to go to 0.  It doesn't quite.  Figure about 200 mV for a small signal transistor in saturation (when it would allow more collector current than the circuit is giving it).  Transistors can make decent switches for many applications, but some minimum on-voltage across the switch will always be there.