Particle classification is very important to keep in mind as you learn more about particles. The particle て discussed in this lesson is a final particle and mustn't be confused with the conjunctive one. Although the conjunctive て may be at the end of a sentence in the instance that the remaining part of a sentence is not said, overall context should always help you differentiate between the two.
The main purpose of the final particle て is to make a light command. This is the contraction of ～てください, the polite command. Likewise, the negative of this is ～ないで（ください）, making the absence of ください a contracted form of the pattern. Thus, its absence for both the affirmative and negative makes it go from polite to plain speech. A more polite form of both can be made by changing ください to くださいませんか.
Please refund this.
Please do your test by closing your textbooks.
Look at this.
Please call a taxi (for me).
Please speak more slowly.
(Do it) faster!
Please go straight.
Please sit on a cushion.
Culture Note: 座布団 are floor cushions used instead of chairs in traditional Japanese rooms.
Contraction Note: ～てて is the contraction of ～ていて. ～ている means "-ing". ～ていて creates a command similar to "be...-ing!".
In women's speech, ～(っ)てよ asserts opinion. However, this phrase has essentially disappeared in the younger generations and is most likely to be used by older women or seen in literature dating back a few decades ago. It is replaced by things such as ～てるよ.
I'm not even the least bit drunk.
From 永すぎた春 by 三島由紀夫.
Particle Note: なんか is essentially a filler word here.14. かまわなくってよ。
～て, with a high intonation, can make a question. These two usages, though, are hackneyed yet refined. This, too, would be replaced with something like ～てる？.15. あなた、私のいうことが分かって？
It is to note that ～てよ is used by everyone to make a command. The sound of your voice is what matters.
～て may also follow things like だ・じゃ to tell someone to do something or give some sort of instruction/warning. This, though, is uncommon.
16. 大変なことじゃて。（Old person; dialectical)
That's a horrible thing!