If it is just one then this wire is commonly referred to as Ground or Return or Common. Next, looks to me like you have your polarity backwards on your example. Vin is a voltage with respect to ground. It is so named because the emitter voltage will follow the base voltage. So if you apply 5 volts to the base you will get 5-.7 or 4.3V at the emitter in this configuration.
Beta is the term used for current gain of a transistor.
There is a DC Beta and an AC Beta, but at the moment just consider DC beta. This allows you to calculate a maximum value for the base resistor. Now what you really want to do is run the transistor in saturatation or cut-off. That is the difference between analog and digital. If you run a transistor in it's linear region and never allow it to go into stauration or cut-off, then you are using it in an analog mode.