5mA to 20mA LEDs are typical for indicator lights. Since efficiency is generally not critical here, it is a good idea simply to use a series resistor as a "ballast" for the LED. If you have a 5V power supply, and the LED turns on with 2V, then the resistor will have 3V across it. 3V / 20mA = 150 Ohm, and this is the value of ballast resistor you'd want. Even if the LED turns on with 1V or 3V, the current will only change to 26mA or 13mA, respectively.
You can't directly place two LEDs in parallel without special engineering; because of their voltage mismatches, one LED will typically hog most (or all) of the current. However, you can confidently place "LED + Ballast Resistor" combinations in parallel.
Typically, the voltage that it takes to turn on an LED will depend primarily on the LED's color, and what technology was used to build it. Infrared LEDs are often ~1V, red LEDs ~1.8V, Yellow LEDs ~2.3V, Green LEDs ~3V, and Blue or White LEDs ~3.5V (white LEDs are just blue/ultraviolet LEDs, which are coated with yellow phosphors to convert some of the light to lower energy). If you attempt to place a 5V power supply directly across any of these LEDs, they will break; you should not feed an LED with a voltage source, but with a current source. In my example above, I used a "ballast resistor" to imitate a current source (an arguably crappy current source, but good enough for these purposes).
How much voltage does an LED take to turn on? How much current can you feed into it? As psurya1994 stated, the details will be in the LEDs datasheet. If you have an LED on your bench and you'd like to find how much voltage it takes to turn on, you can configure your power supply as a 10mA current source (or, if it doesn't have a constant-current option, use a constant voltage power supply with a series resistor) and power the LED, then use a multimeter to read the voltage across it.
Hope this helps.