I just wanted to clarify. I think I have it right but am not 100% sure.
A wizard gets his spell load for the day. Say he gets 2 first level spells and chooses charm and sleep for his spells when he studies.
When he casts, he can cast one of these per round of combat. He only loses the spell if he rolls a natural '2' (AKA snake eyes) right?
Depending on the spell, the target gets a saving throw as per the ability check section.
If the wizard does not study the following day (the party is being chased or has a severe time constraint or something) he loses 1D6 of spell levels. In my example, the wizard would need to roll a 1 to maintain just 1 of his spells and loses them both on a 2-6 for the day until he studies again?
That is mostly correct. If the wizard doesn't study the next day, his spells fade. You roll 1d6 to determine how many levels worth of spells (player's choice) he loses. Highest possible spell levels based on the die roll are lost first.
In this case you want to roll low--if you roll a 1, he loses only one, first level spell.
If, on the other end of the spectrum, you roll a 6, he loses 6 levels worth of spells. If he has any 6th level spells, a 6th level spell is lost first. If he doesn't have sixth, but has fifth, he loses a fifth and a first. If his highest are fourth, he loses a fourth and a second. If his highest are third, he loses a third, a second, and a first. And so on.