Not one stack but several? I'm mixed up

[Patrick <patrick@...>]

So I am making progress learning the C API but there is something 
fundamental I am still mixed up about.

After we create a Lua state most of the functions take a pointer to it:

lua_DoSomething( Lua_State *L )

I assume that this is a complex struct that contains the stack mechanisms.

However rereading this in PIL2, on Page 241:
An important concept here is the stack is not a global structure, each 
function has it's own private local stack

If I push a string onto the stack and then push a number are they not 
sitting on the same data structure? In the Factor language it looks like 
they are but they are in fact not, from what I remember it is some sort 
of overloaded facility. Factors core is in C++, Lua is not, so I don't 
get how it could be overloaded and I don't understand how each function 
has it's own stack but "the stack"(as in singular) gets this value or 
that value if that is the case.

Could someone straighten me out?

Thanks for reading-Patrick