I recorded the last third of my fight against Leviathan and I think it makes for a nice video showing how the game's combat system works.
Setting the stage: the Leviathan fight takes place in a VR simulation. If you win it, you get the Leviathan "summon" materia. Unlike most of the game, your damage and magic power get restored after the battle. Leviathan attacks with magic water spells and is most damaged by lightning attacks - and I came prepared for that. More or less.
The battle starts just as Leviathan is doing one of its massive "tidal wave" attacks, which hits everyone hard, but I had a mana barrier up that cuts damage down.
View: https://youtu.be/glBzHxKY9_U
The green bar (that's green, not red!) is each character's health. The blue bar below is the action (ATB) gauge. Characters can do their normal attack any old time, but special attacks, spells, and using items requires a chunk of the action gauge. The gauge fills up slowly at all times and faster when the character you control is attacking. When the video starts, I'm controlling Cloud and try to do an action, but all options are greyed out because he hasn't quite gotten his ATB high enough. So I switch to Barret, and cast a healing spell on Tifa, then take control of Tifa while he goes through the motions.
When you are picking commands, note that the action doesn't quite stop. At 2:00 in, you can see some shots from Barret hitting and doing damage while I'm trying to decide what to do with Tifa.
Interrupting and dodging are very much things in battle. At 3:20, Barret tries to heal Tifa but he gets nailed by Leviathan's attack while casting so he loses the mana points and action points and Tifa gets no health at all. At 1:30, Tifa tries to hit Leviathan with a lightning spell, but Leviathan dives and Tifa does no damage. This works both ways: your characters can dodge attacks and interrupt, too.
The stagger bar is another big feature - that's the orange bar under the yellow health bar of the enemy. If that fills up, the enemy staggers and falls down. While the enemy is down, you do extra damage.
At 4:55, I get in serious trouble. Leviathan is going to do another tidal wave attack. Unfortunately, Tifa is the one that casts the anti-magic barrier that helped me survive the last one and her ATB isn't close to full. The party's health isn't all that great, either. However, Cloud's limit bar is full. A "limit" attack is a huge attack the character can make on rare occasion and Cloud's adds a lot to the stagger bar. Leviathan's just about out of HP, too. If I can hit it with that limit...
Nope. Cloud starts his big attack, then Leviathan cuts loose and almost wipes the entire party!