How do I use my second move in Pokemon GO?

Note: This answer has been obsoleted since version 0.85.2 (released 2017-12-08).

Please see wondercricket's answer for the new way of using a charge move.

To use a Pokemon's second move, you must press and hold on the Pokemon during a battle. The screen will "shrink" and gain a letterbox effect, and the Pokemon will use its second attack.

However, the second attack can only be used when it's "charged" by using regular attacks repeatedly. During a Gym battle, you can see how charged the attack is by looking at the blue bars.

As you can see by the screenshot in the question, there are two bars. This represents that you can use the second attack twice, provided both bars are charged.

After depleting the bars by using the second attack, you may re-charge them to re-enable the second attack.

The way you use your charged attack has now changed as of 12/8/2017. Previously, you were required to press and hold on your Pokémon to use your charged attack. In this update, Niantic added a new button dedicate to performing charged attacks

Updated the battle interface to include a dedicated button for performing a Charged Attack.

Using an image from Google images, you can see this new button, along with the charged bars which were once on the top left, on the bottom of your screen.