Paradoxically, we never really stopped using THAC0 while in many cases, we never used it in the first place. The only difference between THAC0 and what we do now is that two numbers in the formula switch positions.
In a modern d20 system, (3.x onward) if you want to hit your opponent, you do this:
Roll a randomizer: a d20 in this case
Add modifiers: BAB, attribute modifiers, magic weapon bonuses (usually a plus), buffing/debuffing spells/items, feats (and these are all pretty common, there could be others but they're rare)
Hit a target number: your opponents AC in this case
In the old THAC0 system, if you want to hit an opponent, you do this:
Roll a randomizer: a d20 in this case
Add modifiers: attribute modifiers, magic weapon bonuses, your opponents AC (and these are pretty common, there could be others but they're rare)
Hit a target number: THAC0 (it stands for 'To Hit Armor Class 0' in case anyone forgot)
The only difference in the modern system is that two numbers were swapped. Your old target number (THAC0) is now just a modifier (called BAB). The new target number (opponents AC) used to just be a modifier. And because they switched from descending to ascending AC, your THAC0 number is inverted. So if your THAC0 was 20, your new BAB is +0. If it was 19, then it's +1 etcetera etcetera. You could swap those two numbers back and the game would play exactly the same except that it would be called THAC20 instead of d20.
The problem back in the day was, and this happened even at RPGA events and official cons, was that people would insist on using the opponents AC as the target number. It still works that way, but it isn't how the numbers were presented. No one ever stopped using THAC0, it's just not called that anymore (it's BAB now). The number wouldn't even look different except that they went up with AC instead of down. That's the one minor boost fighter types got in 3.x: in the old editions, everyone started with a THAC0 of 20 (or BAB +0). Now, fighters start with a BAB of +1 (THAC0 19).
It's funny now reading articles and comics about the "death" of THAC0.