Recognizability and complements
Example 1.
The empty language is recognizable and its complement (the language containing all the words) is also recognizable.
Example 2.
Any regular language is recognizable and, since regular languages are closed under complement, its complement is also recognizable.
Example 3.
The language $H = \{ \langle T, x \rangle : T \mbox{ is a Turing Machine}, x \in \{0,1\}^*, T(x) \mbox{ halts}\}$ is recognizable.
In order to recognize $H$ it suffices to build a Turing machine $M$ that checks if  $T$ is a valid description of a Turing machine, simulates $T$ on $x$ until $T(x)$ halts (possibly forever), and accepts.
The complement $\overline{H}$ of $H$, however, is not recognizable as if it was recognizable a Turing machine $M'$ for $\overline{H}$, together with $M$, would allow to solve the Halting problem. To do so simply simulate in parallel $M$ and $M'$ until one of them accepts.