In your example code, gameDay and @"09-01-10" are both examples of NSString objects. In order to compare them, you need to use one of their comparison functions. eg.
Code:
if( [gameDay isEqualToString: @"09-01-10"] ) { ... }
Basically, this is sending one string the -isEqualToString: message, along with another string to compare it to. This returns YES or NO, depending on whether they're equal.
(You could also do [@"09-01-10" isEqualToString: gameDay] ... it's the same difference.)
'=' is the assignment operator. In your first example, you're assigning the address of the object @"09-01-10" to the variable gameDay. (Since this was non-zero, this also evaluated to the same as YES, which is why the code was run.)
'==' is the comparison operator, but it only works on C primitives. In this case, you were comparing the ADDRESSES where the two strings were held in memory, which were different, which is why it was failing.
EDIT: Sorry, BrianSlick, you answered while I was still typing