This entry is when the North American audiences really started to take notice of the series, which had a cult following prior but really wasn't on the same level as anything like Final Fantasy.
Persona 3 is where the series hit its stride with managing relationships to increase rankings in various abilities and in splitting your time between monster raising, battling enemies, and being a high school student in the day. While more accessible than the earlier PS1 entries, Persona 3 is also notable for being quite difficult if you were trying to do everything.
Some of those requests from Elizabeth were incredibly hard, and there were a few boss fights that could easily wipe you out if you weren't holding a guidebook in your hands letting you know which specific Persona to be fusing ahead of time.
What always strikes me most about this entry is the incredibly memorable music that bucked the trend of what you'd hear in a typical JPRG. From the "baby, baby, baby, baby" of the combat music to the operatic take on the Velvet Room, you won't ever forget these tunes. Even the high school hallway theme had an electronic beat you can't get out of your head.
It wasn't all sunshine and gumdrops when Persona 3 was first released however, as the title caused some controversy. Characters point a gun at their foreheads and pull the trigger in a representation of the death of self to release a secondary inner persona. Needless to say, parents weren't stoked about imagery of teenagers repeatedly shooting themselves.