From the posts here, I am able to understand that Daenerys orders Drogon in the High Valyrian language, when she says Dracarys.

The names of the three Dragons are Drogon, Viserion and Rhaegal. Dracarys is High Valyrian for Dragonfire. Daenerys says 'Dracarys' to Drogon to encourage him to breathe fire. So when she said 'Dracarys' in that episode, she was commanding Drogon to burn the slaver.

So, I was curious about how many languages do Daenerys's dragons understand?

Some more languages which they might, are Dothraki and English as Daenerys seem to use them the most throughout the show.

So, are there any examples from the show which show the dragons being talked to, or reacting to some other language apart from High Valyrian?