royal has two "traditional" dining times, which are usually 6 pm and 8:30 pm (though that can change - i was just on liberty two weeks ago and our dining times were 5:30 and 8). it's similar to disney, in that you have the same tablemates and waiters. there are no rotating restaurants, though - you're always in the main dining room.
they also have my-time-dining, or MTD. if you choose that, you can pre-reserve times, or just show up at the dining room when you are hungry. you will have different tables and servers each night. it's basically like eating in a land-based restaurant.
the buffet is also open for dinner (and remains a buffet). and room service is free except for drinks.
fwiw, i have a cruise booked on independence (liberty's sister ship) next february, and for a long time it would not let me reserve late dining. about a month or two ago it finally let me. royal's IT is horrible, so it could just be a website glitch, or for some reason the late dining could be blocked out for the moment but will reopen later. if you're worried about it, i'd call royal; if not, just keep checking occasionally. late dining is rarely full, so you shouldn't have trouble switching into it. worse comes to worst, when you get onboard go see the employee standing in front of the main dining room after you board - they always have someone there to change dining tables, times, etc. You will get a "cruise compass" newsletter when you get onboard (like disney's navigators) and that will tell you when and where you need to go to make changes. i had to do this for my last liberty cruise - even though we had booked late dining, our keys said early seating - and they changed it no problem.
the specialty restaurants on liberty are chops (steakhouse) and portofino (italian). portofino is probably closest to palo. at the next dry dock, rumor is portofino will be changed to giovanni's table, which is also italian, but more casual. i can't remember when the dry dock is, but i think before your cruise. unlike disney, kids are allowed in the specialty restaurants.
sorry, that wasn't short.
but hopefully it helps. i've been on liberty 4 times, so let me know if you have questions.