"I like the mirror."

Translation: Ich mag den Spiegel.

[Jean1707717]

Why did we use " den" here?

[mz147]

Because it's the Akkusativ case, "der Spiegel" as an direct object turns into "den Spiegel". Here, if you like something, that "something" is the direct object of your *liking.
Compare:
Der Mann hat den Spiegel
"The mann has the mirror"
Den Mann hat der Spiegel
"The mirror has the man" (like in the book "Trapped on the other side of the mirror")

[okakeiyou]

how do you identify when to use the 4 cases?

[az_p]

It requires a bit of time to understand until it becomes natural. I recommend looking for explanations on the internet that fit your learning style. Here is one that helped me (in a casual explanation style, but quite a long multi-part explanation): https://yourdailygerman.com/2013/12/06/german-cases-nominative-genitive/