On one circuit I discovered I could connect 3 LEDs in series and they wouldn't need a resistor.
While this might "work" it's a BAD IDEA.*
Here is an [u]online LED resistor calculator[/u]. This is based on [u]Ohm's Law[/u] which defines the relationship between voltage, resistance, and current.
Note that in this case Ohm's Law is applied to the resistor, not the LED. Ohm's Law is a law of nature and it's ALWAYS true, but the resistance of the LED is not constant so we calculate the resistor value from the voltage across the resistor and the current through it.
In a series circuit, the same current flows through all series components and the voltage is divided across the series components. i.e. You might have 5V applied with current of 20mA, with 2V across the LED and 3V across the resistor. (From Ohm's Law, 3V @ 20mA would require a 150 Ohm resistor.)
LEDs are non-linear (like all diodes). As you reach the operating voltage ("breakdown voltage" for a regular diode) the resistance drops dramatically. If you go above the operating voltage you'll suddenly get too much current and potentially burn-out the LED or something else...
LEDs are normally powered by a constant current source, not a constant voltage source. With a constant current source, the voltage across the LED "falls into place". A resistor can approximate a constant current source (if the voltage is constant and high enough).
You can put LEDs in series with the appropriate voltage and the appropriate resistor. As a rule-of thumb, the resistor should "drop" at least as much voltage as the LEDs. A standard LED is usually rated at about 2V, so a 12V power supply with 3 LEDs in series would give you 6V total across the LEDs and 6V across the resistor and that would be OK.
* Some cheap LED flashlights don't have a resistor because they rely on the internal resistance of the battery to limit LED current. That can work with a known battery but if you were to replace the AA cells with D cells, you'd probably fry the LED. And if you try that with a regular power supply, you'll either blow the power supply or the LED. Better flashlights use a switching constant-current regulator, which is more efficient than a resistor and it gives constant light output throughout the life of the battery.