In the following description, is the driving voltage from your microcontroller's output (for example, 3.3V), is the voltage the transistor wants across the base-emitter (usually about 0.6V), is the current the relay needs running through it to be turned on (for example, 100mA), and is the current gain of the transistor (usually about a factor of 100, so if you have 1mA going through the base of the transistor, you can get up to 100 times that, or 100mA through the emitter/collector).
You need to drop a voltage across the resistor, and if your relay needs a current to switch, then your base-emitter current should be (with the factor of 2 as a safety margin, remember the emitter-collector current can only be up to times the base-emitter current and we don't want to design it to be on the edge of just barely working). We have a voltage and a current, so we use Ohm's law to get the resistance: , which we rewrite as and then get . So, punch your components' values in to that formula and you'll get the resistor value to use.
Example: Using the common values stated earlier, we get .