A very thin slice or bit of something is a sliver. A sliver of hope is better than a sliver of doubt, but a having a sliver of wood in your foot is worse. That’s a splinter.
You'll want to watch out for slivers of glass if you break a drinking glass in the sink, and machinists have to wear special protective goggles to guard against flying slivers of metal. If you just want a bite of dessert, you can ask for just a sliver of cheesecake. Sliver is also a verb, meaning "splinter," or "cut into thin pieces." It comes from a verb that's now obsolete, sliven, "to split."