Maybe this will help you.
If you have a typical forward biased transistor, and you start off with no current flowing into the base, initially as you first start injecting current into the base, nothing happens. This is called the “cutoff” region. Then, as you keep increasing the current into the base, you start getting current flowing from the collector to the emitter. While it’s a little non-linear at first, once things really get going, the more current you put into the base, the more current you get proportionally from the collector to the emitter. So in this region, called the “forward active region”, the transistor is acting like a current amplifier. The current from the collector to the emitter is equal to the current going into the base multiplied by the “gain” or “beta” of the transistor. Eventually though, the transistor saturates, and increasing the base current no longer increases the collector-emitter current. This region is called the “saturation” region.
If you are using the transistor as a signal amplifier, you want to bias it’s signal swings so that it stays in the linear part of the forward active region. If you are using the transistor as a digital switch type thing with only an on or off state, then you want to drive it all the way to cutoff when it’s off and all the way to saturation when it’s on.
Relating this back to your arduino, what you are typically going to want to do is connect the base of the transistor to the aduino through a resistor. The value of this base resistor isn’t critical. It has to be small enough that it allows enough current to flow when the arduino turns on to drive the transistor completely into saturation, but the resistor has to be large enough that it also prevents too much base current, which would damage the transistor. You’ll connect the transistor’s emitter to ground (common), then connect the device that you are turning on and off between the collector and the device’s positive voltage supply, often with a current limiting resistor in series with the device. When the arduino output turns “on”, then current will flow into the base, turning the transistor on, and current flows through the device, through the transistor (from the collector to the emitter), then into your “ground” (common). If you are measuring voltages, when the arduino output is off, it’s voltage will be low, and the transistor’s collector will float high since the transistor is basically switched off and it’s in a high impedance state. When the arduino turns on, it’s output will be high, the transistor will turn on, and the collector voltage will drop to a low value since the transistor is now in a low impedance state.
When you use a transistor like a switch, keep in mind that it’s not really a switch. It’s either high impedance or low impedance, not on or off. Transistors never switch completely on and become a conductor like a piece of copper, and transistors never switch completely off. They always leak. This can be an issue if you are driving something that requires very little current. The leakage from the transistor may be big enough that it turns the device “on” even when the transistor is off.
I hope this is helpful to you.