To give a really brief description, Persona (3 and 4, not 1 or 2) is like a mix of a normal JRPG, Pokemon, and a dating sim.
In more detail: The game is sort of divided into two parts, the dungeon-exploring monster killing boss defeating part, and the going to school and making friends part.
In the RPG part you fight by summoning various Personas. All party members have Personas, but the main character has the unique ability to switch between Personas, each of which have different stats and abilities.
You get new Personas one of two ways: First, you can win very basic Personas after battles, but mostly these are just fodder for the second method, which is fusion. Each Persona is assigned a Tarot Arcana, and when you fuse them the resulting Persona is based on the Arcanas and base levels of the Persona that went into it.
Fusion brings us into the importance of the school life part of the game. At school (and also around town) there are a number of people you can be friends with, some of which can also become romantic relationships. All of these friendships, called Social Links (or S. Links) is also assigned a Tarot Arcana, and as you become closer with the person the S. Link ranks up, to max rank of 10. At rank 10 you will unlock the ultimate Persona of that Arcana.
In addition to that, when fusing a Persona, it will recieve an experience bonus based on the rank of the S. Link.
Like a dating sim, there is only a limited amount of time in which you can complete all the S. Links, many of which are only available on certain days, so it becomes a matter of time management to make sure you can finish all the Links while also improving your 'social stats' as I call them (I forget the exact names but it's something like Intelligence, Courage, Charisma) which are needed to make certain choices and start certain S. Links. Over the course of each S. Link you will need to make a number of choices, each choice having a different impact on how much your relationship improves (or can even damage it if you say something totally wrong). Each S. Link has its own story arc that is (usually) thematically connected with the meaning of their Arcana in Tarot and the main plot.
So why are people so into Persona? A few reasons: First, as described above, the fun and innovative (and VERY challenging) gameplay. Second, it has a great story, great writing with equal servings of hilarity and tragedy, great characters. Third it has some of the best music in any video game (imo). For a few good examples check out Mass Destruction, the battle theme of P3, Battle for Everyone's Souls, the final boss theme, also check out I'll Face Myself (Battle) the boss theme of P4.
If you have a PS3 I think both P3 and P4 are only $10-$20 (not sure how many pounds) on PSN, and unless what I've described above sounds absolutely dreadful, it's definitely worth trying out.