Learn French: Je n'aime pas le matin. - I do not like the morning.

When the verb starts with a vowel, like aime, the negative word ne drops its e and attaches to the verb with an apostrophe. This makes the sentence flow better when speaking. So, ne aime pas becomes n'aime pas.

Does this sentence mean I don't like this specific morning, or mornings in general?

It usually means you do not like mornings in general. In French, when you express likes and dislikes with verbs like aimer (to like) or détester (to hate), you always use the definite article (le, la, or les). So le matin here translates best as mornings.

Since we are learning about time, if I want to say it is 8 in the morning, do I say huit heures le matin?

No, when you attach the time of day to a specific hour (like saying a.m. in English), you use du matin. For example, Il est huit heures du matin means It is eight in the morning. You use le matin when talking about the morning as a standalone part of the day.