Don't panic. Though probably by the time you have read this, you have already tried a few things that haven't worked, for next time try not to panic and pull at the splinter (unless plenty of it is showing).
Set a $50 bill on the table. 2-20s and a 10 work, but the 50 is best. This is your copay. This, plus 4 hours in an ER, is your motivation.
Take a medically appropriate amount of ibuprofen based on the label information. This will help decrease swelling in the area, hopefully making it easier to pull the splinter out.
Get a high-quality pair of tweezers. Don't use your fingernails, a knife, a needle, or another sub-par instrument to try to dislodge it. Even if you have to wait for an hour or so to go buy them, get a good pair ($10+ at a drug store).
Get out your fingernail clippers.
Get in good, bright lighting.
Clip the nail back as far as you can, but don't clip through the splinter. To do this, clip on one side of the splinter so that a flap of nail is created (i.e. don't separate the piece from the rest of the nail). Pull this flap across to rip the nail portion that is over the splinter. Once you have cleared the splinter, clip the nail from the other side to remove the piece. Keep doing this until you are all the way to where the skin is attached to the underside of the nail.
Remember: this should have exposed a bit of the splinter. Get your high-quality tweezers and keeping them open plenty wide to stay on both sides of the splinter, push and wiggle them under the nail as far as you can. Remember that $50 on the table.
Close the jaws and pull the splinter out. Keep working at it, unless you are tearing away at the exposed splinter. If this happens, you may want to try to dig at it with a sterilized needle or get a better pair of tweezers.
Get some wood or Elmer's glue if this doesn't work. Apply a small amount to the head of the splinter and wait for it to soak into the wood and harden. Carefully peel the glue off and pull it straight out. It may pull the splinter out with it.
If that does not work, go to the pharmacy and get Ichthammol (drawing slave). Next, apply a decent amount of the Ichthammol onto the front of the nail, near the splinter, and put a band-aid on it. The next day, peel off the band-aid and the splinter should be close enough to pull it the rest of the way out. Ichthammol does smell like tar, like used on roadways, but it may work.
Stay calm. If this does not work, you may have to go to the ER, or, if the pain isn't too great you can let your skin naturally expel it or the fingernail grow out, which will push it out.