Simplest Example of a Poset that is not a Lattice

The set $\{x,y\}$ in which $x$ and $y$ are incomparable is a poset that is not a lattice, since $x$ and $y$ have neither a common lower nor common upper bound. (In fact, this is the simplest such example.)

If you want a slightly less silly example, take the collection $\{\emptyset, \{0\}, \{1\}\}$ ordered by inclusion. This is a poset, but not a lattice since $\{0\}$ and $\{1\}$ have no common upper bound.

(The previous answers are perfectly fine, but it's always helpful to have a picture in mind.)

In the DAG of a lattice every pair of elements must have both a common successor (and thus a successor is the sup) and predecessor (inf).

Non-examples: no common predecessor for 1 and 2:

  3
 / \
1   2

No common successor for 2 and 3:

2   3
 \ /
  1

No common successor nor predecessor for 1 and 2:

1   2

Being a lattice implies that the poset is connected, and the above is not connected.