Understanding LED specifications (hobby kit)
rated current is the current level used to determine the lifetime of the led and it's brightness,  20mA is fairly common for that style of LED. 

fusion current is the amount of current that will cause immediate failure. 
For that type of led somewhere around 200mA but anything above the rated current will cause accelerated ageing. Exceeding the rated current is done at your own risk.

The minimum current before it  will show light is forward leakage current. 

With modern LEDs in a dark room fractions of a microampere will cause a perceptable glow. so forward leakage current is close enough to zero for
most applications, and it is often not quoted on data sheets.

Another parameter lumens per miliampere will tell you how much current will get how much glow. often this is only given at one current level.
eg "87 lm @ 20 mA" but this relationship is essentially linear.