What exactly is a lattice? And can somebody give an example of something that is not one?

Question

Looking at the (very brief) definition in my textbook with no examples, I have the following:

A poset $(A,\preceq)$ in which every two elements have a greatest lower bound in $A$ and a least upper bound in $A$ is called a lattice.

But I can't think of a poset that doesn't have a GLB and LUB...

Answer 1

What about this one? What’s the least upper bound of the two fellows at the top?

        *   *  
         \ /  
          *

Answer 2

$$\Huge\ldots\vphantom{Some filler, if only there was a two-dots symbol}$$

Answer 3

I (and presumably William of Ockham) suggest a 2-element antichain.