BJTs amplify current.
In the common emitter configuration, a small base current controls a larger collector current.
Base current * the transistors gain is how much collector current 'could' flow.
The collector resistor determines the maximum current that can flow through the collector.
If the base current was 1mA and the gain was 100 a collector current of 100mA 'could' flow.
If your collector relay coil was 100 ohms and the supply voltage was 5 volts, (5 - .2Vsat)/100R = 48mA.
i.e. the transistor would work like a switch and 48mA load current would be allowed.
So you have 1mA of input current controlling 48 mA of output current, this is a current gain.
To get a base current of 1mA from a 5V Arduino: (5V - .7Vbe) / 1mA = 4.3K for the base resistor.