Is this circuit acting as an amplifier and how does it work?
What you have is called diode connection configuration.  Basically, Vb is always a diode voltage drop higher than Ve.  Shorting Vc to Vb only helps with biasing the transistor.

Your circuit is equivalent to this:







simulate this circuit &ndash; Schematic created using CircuitLab

Your transistor acts as a diode.  This kind of configuration can be used in current mirrors when you want a forward voltage drop from a diode to match the drop from another transistor such as in Wilson current source.  But it is useful only from transistors on same die.

You can also use this to replace a diode in a circuit if you will.  At low voltage the voltage drop is usually quite small.  Yet, you are better off using a proper diode unless this is part of a much more complex circuit where such an arrangement is needed (eg. wilson current source).