With the iPhone 6 the slot is even smaller and narrower making this even more fun.
The best thing to do is to take it to an Apple Genius Bar.
Do NOT try this next part. You will likely damage your iPhone. It's stupid. Don't do it. You're compounding an already stupid mistake to try this.
But here's what I did:
I could see the edge of the SIM card.
I took a pin (a pin, not a needle, the are harder and tend to break) and bent the end into a right-angle. So there was about 4mm of it at an angle.
I took a drop of Superglue (Stop reading. Go to the Genius Bar, now). and placed it on some paper (away from the precious phone!).
I dipped the end part of the needle into it. And waited. I waited about 30 seconds. I dipped it again until it was getting tacky (reduced chance of dripping into the iPhone, and also a reasonable coating of glue).
I carefully put the pin into the SIM slot and guessed that the pin was along the edge of the stuck SIM.
I pressed a little.
I stayed very, very still.
I waited.
I waited for two minutes.
I let go of the pin and then very, very slowly dragged it directly (not at any kind of angle) out of the iPhone, with the SIM attached.
Really. Go to the Genius Bar. Don't listen to crazy Internet people talking about putting Superglue coated things into your iPhone.