I misstated it somewhat, yes. The $90 gets lent out--and where does it end up? In someone else's bank deposit. Then $81 of that gets lent out, and ends up--in yet another bank deposit. And so on...
At the end of this process, your original deposit has been multiplied by the reciprocal of whatever the reserve requirement is. So a 10% reserve means a multiplier of 1/.01 = 10. So your original deposit of $100 turns into a total of $1000. (So I shouldn't have said that your $100 turns into an additional $1000 lent out--I should have said the total money supply is $1000.)
Remember also that, even though $90 of your $100 was lent out, you can still walk into the bank and withdraw the $100 any time; you don't have to wait until whoever borrowed the $90 pays it back. The same is true of all the other deposits spawned by the $90 that is borrowed, and the $81 borrowed against that, and so on... So, again, the final effect of your depositing $100 is that the total money supply is $1000.