The two specs you need are the voltage drop across each LED and the current required to light it to the brightness you want it to show.
Typical red, orange, amberm yellow and green LED's drop around 2.5 - 2.8 volts. Blue LED's are closer to 4 volts, each. Your power supply must provide voltage equal to the voltage drop across each LED times the number of LED's in the string plus the drop across the circuitry used to set the current in the string, which can be a resistor or more complex transistor circuit.
For example, if you have red LED's with a typical voltage drop and a 12 volt supply, you could string 4 LED's in series.
12 volts / 2.5 volts per LED = 4 LED's leaving 2 volts available for the current setting circuit.
If each of these LED's requires 5 mA (.005 A) for the brightness you want to show, you can calculate the resistor value by dividing the 2 volt drop across it by the current through it.
2 volts / .005 mA = 400 ohms.
There are limits to the ways you can do this. For example, LED's of different colors, or even different models of the same colored LED from the same manufacturer, may have different voltage drops or different efficiencies (light output / at a given current). This means, for even color and light output, you should use the same type of LED for each string.
If your power supply isn't regulated to provide constant voltage at any current load, you can make a simple current setting circuit from as few parts as a 50 cent transistor, two resistors and a diode. This circuit would give you constant light output over a wide range of supply voltage, as long as the total voltage doesn't exceed the ratings of the components.
See this article for more info about current sources.