Let's start this.
Original Fiction.
I'll be honest, I don't write that much original fiction. I get way too distracted easily. Nonetheless, I have tried, and my NaNo Novel is actually an OF. So what makes this such a great story type? I'll tell you why.
You're free to make your own world. Like, this is THE best part about OF, at least in my own opinion. I mean, it makes writing a lot easier, but also a lot harder and time consuming at the same time. You're able to make up your own rules and do what you'd like, but you also have to plan it all out so you don't have conflict in it (like, for example, the law. You can't have one rule that allows something and another rule that kinda interferes with that, you know?).
The characters are your own. Don't like annoying/whiny characters in a story? Good. Kick them the hell out of yours! Or maybe you don't like the ones that are snobby or weak and helpless or something. Kick them out, too! EVERYONE GETS LE BOOT! Creating characters is a challenge because you want them to be developed (I don't think that was the word I was originally going for, but hopefully you know what word I'm talking about...), but want them to appear human, too. You can make them as kind as the Good Witch or as evil as Dr. Evil. They're yours for the taking!
Plotline! You're gonna own this. You get to make up the story. Who doesn't like making up their own unique story? You could go anywhere with it! And it will be your baby.
Fan-Fiction.
I write a ton of Fan-Fiction and have been writing it since I was 12 (so for eight years now). It's my favorite type to write and so I will now tell you what I believe makes this a great story type, too.
It's an escape. You must be thinking, Well, original fiction's an escape, too! I agree. You're right. But for me, personally, FF is an even bigger escape. Let's actually add that to the list for OF. It's an escape as well. Getting back to FF... it's a great escape for me because, while I like reading some OF to forget my daily troubles, Fan Fiction gives me that familiarity that I sometimes like. Which leads me to my next few reasons...
The Characters and the World. I said I like familiarity in FF, and the biggest reason is the characters and the world they live in. I love reading about them and their little quirks. Call me weird, but I think the best fan fiction stories are the ones that don't necessarily have the best plot, but the ones with the most realistic characters. I mean, most people write for the characters and when you're able to write scenarios and the reactions from the characters you're portraying, it's just amazing. Like... it's spectacular. Then we have the world which is just as important and amazing. For example, Hogwarts. It's magic and some people have been able to connect with the Harry Potter series on such a level that it's just like home to them. Some people read FF just for the world alone.
The OCs. (This was probably inevitable.) Some people tend to forget that not every original character is a Mary Sue. In actuality, I think there are far less Mary Sues than people think in some FF stories. What really constitutes a Mary Sue? An author putting his or herself as the main character, right? Well, that's a bunch of crap to me. I feel like nowadays people can't have a humble, kind character without the audience believing it's a Mary Sue. When authors are able to create a good character that's relatable and likable, I applaud the author. Even when they create the most evil character ever I'm just blown away. Some FF writers take the time and put in the effort to really make their OCs stand out and shine. I think that when you're able to create an amazing character for your Fan Fiction story, it just makes the world a whole lot brighter.
I'm going to end this here as it could get longer. I just wanted to give you some brief examples as to why I think FF and OF are amazing story types and I hope I've made my point. Let me know what some of your favorite parts about these types are. :)