If you want to tell someone to stop copying you, you can say:

マネしないでよ!
マネ　しないで　よ！
Don't copy me! // Quit copying me!
Literally: "mimicry / imitation + don't do + よ！"

The key word here is マネ, which is sometimes written in kanji as 真似（まね）and means "mimicry" or "imitation."

It is easily confused with the word フリ, which is sometimes written in kanji as 振り（ふり）and means "pretending," "show," or "behavior."

One more time, that's:

マネ＝copying (someone)
フリ＝pretending (to be [someone/something])

As we saw above with マネ, we can also attach する to フリ, like this:

赤ちゃんが泣いたフリしてる。
あかちゃん　が　ないた　ふり　してる。
The baby's (just) pretending to cry.
Literally: "baby + が + cried + pretending / show + is doing."
Note: I'm not totally sure why we have "to cry" in the past tense, 泣いた, even though the baby is crying right now. That's just the way we say this kind of thing.

So that baby is not really crying. He is pretending to cry.

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真似 and 振り are both nouns. You'll often see them coming after:
✔ Verbs in plain form.
✔ The particle の.

Although that's not the only way you'll see them. For example, here is マネ coming after the particle を：

漫画のセリフを真似する。
まんが　の　せりふ　を　まね　する。
He copies lines from manga.
Literally: "manga + の + lines + を + mimicry / imitation + does."

On a side note, you may wish to refrain from mimicking manga and anime characters while in Japan, especially on a first date. Unless it's with your waifu... in which case, we may have some other issues to deal with.

On a second side note, sometimes if I do something stupid, Rei will copy what I'm doing and say:

見て、ニコのマネ。
みて、ニコ　の　マネ。
Look, I'm Niko (doing something stupid).
Literally: "look, + Niko + の + mimicry / imitation."

If you want to level up your マネ to a full-on impersonation (e.g. of a famous person or character), then you can say モノマネ, like this:

あたしジャックバウアーのモノマネできるよ。
あたし　じゃっくばうあー　の　ものまね　できる　よ。
I can do a (great) Jack Bauer impersonation.
Literally: "I + Jack Bauer + の + impersonation + can do + よ."

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💀 Quiz of Doom! 💀

You're (a male) going to a wedding.

Your demon ex-girlfriend is going to be there, and there is a good chance that she'll attempt to lure you into the depths of relationship hell again. So you ask your badass friend (who just so happens to be a cool, beautiful, well-spoken girl) to be your date, even though you're just friends.

On the way to the wedding, you want to say:

"Will you pretend to be my girlfriend today?"

Question #1: Do you use マネ or フリ？
Question #2: How do you say that sentence in Japanese?

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Answer time!

Answer #1: Use フリ！ Because she's not really your girlfriend--she'spretending.
Answer #2:

今日彼女のふりしてくれない？
きょう　かのじょ　の　ふり　して　くれない？　
Will you pretend to be my girlfriend today?
Literally: "today + girlfriend + の + pretending / show + do + will you not give me?"

By the way, there are surely other ways to say that. So don't fret if your guess wasn't exactly the same.

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To wrap up, here are a few more examples...

ジュリアの真似してショートカットにしたんだ。
じゅりあ　の　まね　して　しょーとかっと　に　した　んだ。
I cut my hair short just like Julia's.
Literally: "Julia + の + mimicry / imitation + do + short cut + に + did + んだ."

英語全然聞き取れなくて、ずっと分かってる振りしてたよ！
えいご　ぜんぜん　ききとれなくて、　ずっと　わかってる　ふり　してたよ！
I couldn't catch their English at all, so I was just pretended to understand the whole time!
Literally: "English + not at all + cannot catch / follow, + the whole time + am understanding + pretending / show + was doing + よ！."

寝たフリしてるのバレバレだよ。
ねた　ふり　してる　の　ばればれ　だよ。
It's totally obvious that you're just pretending to be asleep.
Literally: "slept + pretending / show + are doing + の + totally obvious + だよ."
Note: バレバレ comes from the verb ばれる, "to leak out (a secret); to be exposed (when lying, cheating, etc.)." You can say バレバレだよ anytime someone fails miserably at hiding something.

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Grammar tangent!

The first half of that can be a sentence on its own:

寝たフリしてる。
ねた　フリ　してる。
(He's) pretending to be asleep.

We nominalize that sentence (i.e. make it into a noun) by sticking a の onto the end of it.

Then we connect it to the second sentence, which is:

バレバレだよ。
ばればれ　だ　よ。
(It) is totally obvious (that you're lying, cheating, etc.).

So when we say:

寝たフリしてるのバレバレだよ。
ねた　ふり　してる　の　ばればれ　だよ。
It's totally obvious that you're just pretending to be asleep.

...it's almost like we're saying:

寝たフリしてるの //（は）// バレバレ // だ // よ。
The fact that you are pretending to be sleeping //（は）// exposed // is // よ.

Hope that wasn't too confusing. ^_^