How do you come up with ideas?

[Nylan]

Not the best method by any means, but good for overcoming writer's block:

1) Go to http://tvtropes.org
2) Press the "random trope" button
3) Do you like this trope? --> add it to your story
Do you want to subvert this trope --> think of a fun way to do so in a story
4) rinse and repeat as desired

Everything has been done before. The question is, what do YOU want to tell the world?

[Amecha]

Multiple ways.
I've taken dreams and nightmares and made them into stories. I had one particular dream about a girl with a book she could open and pull anything out of (including entire rooms and her house). I'm not 100% sure if I will end up using it as a plot but its always something I could use someday. Nightmares are great too even if you don't keep the scary aspects, or maybe you do a make a horror. I have an idea brewing in my head around one of my nightmares from years ago. I also had a dream about a group of people standing on a beach near a shack and it managed to turn into a huge fantasy plot. I only take a detail or two from the plot and warp a story around it.
Another idea is straight taking ideas from other works. I read a manga years ago that was enjoyable enough and I like the plot enough but I felt a little irritated at an opportunity they'd missed with the story and a single "what if" spawned this huge story that I am still working on to this day and intend to make a graphic novel about. Originally it was pretty similar to the plot of the manga but now its so different you couldn't even compare the two.
Another way I do it is by sitting down and making a character. As you design them just adding and playing with features and traits you need to start thinking about why they have those features and traits. I made a character once that was just a cookie cutter antagonist and was merely a plot device, while considering their haircut I started doodling and suddenly I had this weird haircut that tied into the character's relationship with their parents and their childhood and suddenly he was someone I HAD to tell a story about. It practically wrote itself.

Just find an anchor. Anything can be a story. Even writing prompts can make a solid story. Pick something to start with and start small, very small. I've never succeed in making a plot I though was interesting by going "I want an epic fantasy trilogy" and trying to force it, but I've made plots that would be an epic fantasy trilogy by going "what if".

[Shinoki]

Take things you've seen or interesting events that have happened in your life and simplify them down to a concept and then start elaborating there.

I personally have a bunch of random dreams and use them for inspiration. Half of them are self-insert fanfic where I'm the self insert, but they have some pretty unique stories when you look past the messy randomness of dreams.