Q:

When did dilithium crystals become so important for starships? 

Every Star Trek fan knows that dilithium crystals help power the Enterprise (at least Kirk's Enterprise and later), but at what point did they become necessary or widely used in the Star Trek history? 
I don't think Zefram Cochrane had any dilithium laying around when he build his first warp ship in 2063, and I don't recall mentions of it during Captain Archer's time (but I could be mistaken). 

A:

From Memory Alpha: Constitution-class starships continued to use crystallized lithium until the mid-2260s, before switching to dilithium. 
The article is pretty detailed, and if you're looking for more specific information, I'd take a look. 

A:

Dilithium is a crystal used to regulate the matter/anti-matter reactions in the core.  But they are found on very few planets, although it was eventually possible to create it artificially and cheaply.  Cochrane used an old Titan missile for his first flight, with no obvious sources of dilithium around.  However, since dilithium is simply used to regulate the reaction, it seems likely that he just used a different way to regulate it.  Perhaps dilithium is just a more efficient way of regulating this reaction, which is why its used so much in the future.