We call them "pin" feathers because they look like tiny spiky pins (especially when on the head) on our birds. Baby birds have loads of them, but even an adult bird who is going through a heavy molt can have that spiky look too.
Pin feathers are simply new feathers coming in. Pin feathers have a sheath of keratin surrounding them which helps support and protect the new feather as it grows. Depending on the type of feathers growing, they can be quite sensitive sometimes and your bird may not want you to touch them when they are first coming in. If they are a new wing or tail feather, they are often referred to as blood feathers, and they are usually the most tender when first coming in. They do have a blood supply which nourishes the feather as it grows and if accidentally broken or damaged, your bird may bleed profusely, so you always want to take and help your bird protect these larger incoming feathers. Once the feather begins to grow out and mature, the blood supply will begin to recede and when the feather is fully formed, the tenderness is also gone.
Many birds enjoy you gently rubbing pin feathers on the head and cheek area to help them remove the feather sheath so the feather can open completely. Warm baths also sometimes help the feather sheaths soften and come off more easily.