My teacher told me that the word "Run!" is a sentence. Can anyone explain why this is considered a sentence? Does it mean any action verb can become a sentence? Like dance, eat, or jump?
This is known as the imperative mood, which is used to give commands. With English language imperative sentences, the subject is implied to be "you". We don't need to write or say the subject in this case because it's already clear that commands are directed to "you". In contrast, in the indicative mood (describing something, not commanding), many different people might be the runner, so you need to specify who.
Once we understand that the subject is "you" in this case, it is easy to see that the sentence is complete: it has a subject, a verb, and the idea (you're being commanded to run) is complete.
As an analogy, in many other languages, the subject can be dropped without losing any context about who is "doing" the verb. For instance, in Spanish, one might say "corres" instead of "tú corres" for the indicative "you run". It's much rarer to see this occur in English, but imperative mood is the prime example of it.
[You] Run! (a command to run)
You run. (an 'indicative' mood statement describing your action)