A needle is a small, pointed piece of metal that's used in medicine or sewing. When a tailor sews a button on your shirt or even when a doctor sews your skin back together, a needle is used.
Needles are sharp — when you stitch fabric with a needle, you might use a metal thimble to protect your fingertip. When a doctor or nurse uses a needle, it needs to be very sharp — it has to be able to penetrate human skin to stitch up a wound or inject a medication. When you needle someone, you don't literally poke them, but you do annoy or provoke them in some way.