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Can somebody tell me what $\mathbb{Z}_3$ means?

Does it mean the positive integers up to $3$?

$\mathbb{Z}_3\stackrel{?}{=}\{1,2,3\}$.

Or is it something else?

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    In what context? It often means ``the group of integers mod $3$." – 2017-02-13
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    Notation rarely means just one thing in all contexts. So can you give us some context? Where did you come across this notation? – 2017-02-13
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    It depends on the context. Usually, it means $\Bbb Z /3 \Bbb Z$, that is, the integers modulo 3. Sometimes, it refers to the 3-adic numbers – 2017-02-13
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    Also might be worth mentioning that "the integers mod 3" are (0, 1, 2) and not (1, 2, 3) – 2017-02-13
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    @Sunsevn technically both are correct, with $3$ denoting the additive identity. – 2017-02-13
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    And sometimes, it just means the set $\Bbb Z_3=\{1,2,3\}$. – 2017-02-13
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    @Omnomnomnom: Really? I don't think I've seen that notation. I've seen $[3]$, but not $\Bbb Z_3$. – 2017-02-13
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    @tomasz it's rare, but I've seen it on this site. Usually in the context of set theory/combinatorics – 2017-02-13

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It may mean either the field with $3$ elements $\mathbf Z/3\mathbf Z$ or the ring of $3$-adic integers, i.e. $\;\biggl\{\sum\limits_{k\ge0}a_k 3^k\:\Big \vert\: \forall k,\;0\le a_k\le 2\biggr\}$.

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    @mr_e_man: Oh! yes/ Thanks for pointing the typo. – 2018-07-30