A very good installment in the Final fantasy series!
That out of the way, I suppose I'll start with the story of the game. There are two worlds in FFXIII, those being Cocoon and Gran Pulse. People on Cocoon fear Pulse, stating it is Hell, pretty much. Another thing people fear are l'cie, since they are associated with Pulse. a l'cie is a person who has been branded by a Fal'cie and must do some task for said fal'cie. Complete it, you get a nice crystal slumber, which is basically dying in a crystal (Woohoo, thanks Fal'cie >_>). BUT if you do not complete your task, you will eventually become cie'th, a monster, a grotesque abomination, and no one likes to be a grotesque abomination.
The two characters you start with are Lightning AKA Cloud souped on estrogen and Sazh AKA Mr. Chocofro. And then you start battling through soldiers and learning the very complex ropes of this game. After a bit of of kicking ass with Ms. Cloud with less hair gel, you get introduced to Snow AKA THE HERO, and you start whooping some ass with him and his gang. Then after our good ol' hero reaches a group of defenseless civilians, we see Hope AKA whiny contradictory name and Vanille AKA Yuffie/Selphie/Rikku/Penelo. later on in the Game you get introduced to Fang AKA...uhhh...I can't think of any cute nicknames for her, but still, you see Fang. And there you have it, your six heroes of FFXIII! For the most part they are good characters. Lightning is cool, Snow I actually find tolerable and I like him (though some hate him, even though he's a HERO, crazy I know...), Vanille is.....meh, Hope is annoying at first, but he grows into a not whiny little prick later on, Fang is pretty damn awesome, and Sazh, oh man, sazh, he is freaking AWESOME. His backstory, his personailty, his ****ing chocofro. You know Sazh is cool, since chocobos would never live in the hair of an uncool guy!
Overall, I like the story, more for the characters than the plot, though the plot is fine, too. Dialogue can be pretty silly at times, but I personally can overlook that. There were even a few events in the game that manage to get to my emotions, but I could just be a sappy wimp, but for the sake of my ego, I'll just say Square did the do a good job invoking emotion in their scenes (I'm still a manly man, dammit!). The story kinda loses some the spotlight later on in the game, though, by disc 3 (in terms of the 360, anyways; the PS3 version only uses one disc), and gameplay starts hogging some of that light by then.Speaking of which, let's talk about Gameplay.
The main point of battling in FFXIII are paradigms, where you set different roles to characters, and customize different sets of character roles that you switch between in battle to fit the situation. You got Commandos, who do physical damage and have your allies attack whoever the commando attacks, ravagers who do magical damage, medic who focus on healing, saboteurs who debuff the enemies, synergists who buff your party up, and sentinels, who provoke enemies and then defend and take damage to help keep your other two characters alive and unoccupied. This system allows for alot of strategy, such as which roles should you rely on for the battle? How should you group together roles? Which character should be which roles? You will certainly be switching between paradigms plenty during boss fights, though during many early enemy fights you probably won't as much. Later on when you get around disc 3, however, things get much more difficult, so plan on switching paradigms more (or maybe not and I just suck...nah, that can't be it. Silly me). At the end of each battle you get a rating out of 5 stars; the faster you win, the more stars you get, and the higher the stars the more likely you are to get some drops from enemies (I believe), which are needed to upgrade weapons and accessories, and to sell for Gil (money), since that's pretty much the only way to get Gil in the game. You also get introduced to TP (technique points) at some point in the game, which allow you to use libra, summon, and to use some especial abilities.
Also, the summons in this game are referred to as eidolins, which you acquire by beating them in battle a certain method. each character has their own individual eidolin that you can only summon if you control that character and have enough TP. Like FFVI, FFVIII, and FFX, summons in this game actually serve a purpose, and aren't just spells with longer animations, or are just plain useless like in FFXII. in FFVI and FFVIII, summons could be used to increase stats and gain new abilities, and could be used to damage foes, too, though XIII is more like FFX, where the summoned beast actually stay on the battlefield to do multiple attacks, and keep your whole party from dying! Ain't that just nice of them?
Outside of battle you get to tinker around with the crysterium, which is pretty much a prettier, shinier version of FFX's Sphere Grid and FFXII's LP Grid.You increase stats, gain new abilities, gain accessory slots, and gain ATB (extended ATB bar for extra attacks per turn) via the crysterium. In order to advance the crysterium, you need CP (crystal points, which you gain after every battle you face (except eidolin battles, I believe). You can also upgrade weapons with enemy drops or store bought materials to increase the stats that come with said weapon, but all this will do will make players hesitant to try out other weapons when they already have an upgraded version of another weapon that has better stats of a not upgraded weapon (but the not upgraded weapon could actuallly be better in the long run after it gets upgraded, UGH!). Unless you do research online on what weapons are best for each character, methinks you will go through this.
Overall, the gameplay is nice and I enjoy it, though for SOME people it may not get good until Much later in game (SOME being the key word there, I enjoyed the gameplay in it's beginning parts, though can see how others may not). And another thing FF fans seem to dislike is how FFXIII is so linear, which it is, but I, being a fan of FFX which is also quite linear, don't see it as an issue. And also, you don't get to switch between party members early on in the game; the game forces you to use characters, but me also liking FFIV, do not mind this either.
And finally onto presentation. The graphics, of course, are quite tremendous, and I loved them alot, and I only played the 360 version! The PS3 version is supposed to look a bit better, so I can only imagine how good the PS3 version looks. The music is quite nice in this game, and there are some very memorable songs.
So, looking at it all, it has great, strategic gameplay once you get used to it, a good story, great looks and music, and actually implements summons well! BUT gameplay takes a while to really shine, and when it does shine, the story shines less, an upgrade system I did not care for at all, and some not-so-award-winning dialogue to be heard sometimes. And some characters may be intolerable at times.