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Do you know any example that demonstrated Zorn's Lemma simple and intuitive? I know some applications of it, but in these proof I find the application of Zorn Lemma not very intuitive.

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Many of the most straightforward applications of Zorn’s lemma can be seen as slightly disguised applications of the Teichmüller-Tukey lemma. I doubt that you can make either very intuitive, but the special case of sets of finite character seems a little easier to recognize and get a feel for than the general case. –  Brian M. Scott Dec 5 '12 at 23:33
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Zorn's lemma is not intuitive. It only becomes intuitive when you get comfortable with it and take it for granted. The problem is that Zorn's lemma is not counterintuitive either. It's just is.

The idea is that if every chain has an upper bound, then there is a maximal element. I think that the most intuitive usage is the proof that every chain can be extended to a maximal chain.

Let $(P,\leq)$ be a partially ordered set and $C$ a chain in $P$. Let $T=\{D\subseteq P\mid C\subseteq D\text{ is a chain}\}$. Then $(T,\subseteq)$ has the Zorn property, because a chain in $T$ is an increasing collection of chains. The $\subseteq$-increasing union of chains is a chain as well, so there is an upper bound. By Zorn there is a maximal element, and it is a maximal chain by definition.

If you search on this site "Zorn's lemma" you can find more than a handful examples explaining slightly more in details several discussions and other applications of Zorn's lemma. Here is a quick list from what I found:

  1. Is there any motivation for Zorn's Lemma?
  2. Every Hilbert space has an orthonomal basis - using Zorn's Lemma
  3. How does this statement on posets follow from Zorn's lemma?
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It is not intuitive. It's not without reason that this joke is well known:

"The Axiom of Choice is obviously true, the well-ordering principle obviously false, and who can tell about Zorn's lemma?" — Jerry Bona

I defer to K. Conrad's "blurb" about Zorn's lemma: http://www.math.uconn.edu/~kconrad/blurbs/zorn1.pdf:

Zorn's lemma provides no mechanism to find a maximal element whose existence it asserts. It also says nothing about how many maximal elements there are. Usually [...] there are many maximal elements.

In particular note that the upper bound created during a proof using Zorn's Lemma does not have to be a maximal element. As K. Conrad remarks in Example 2.3, the ideals $\{6\mathbb Z, 12\mathbb Z, 24\mathbb Z\}$ in $\mathbb Z$ have $6\mathbb Z$ as an upper bound (with the partial ordering just being set inclusion), but $6\mathbb Z$ is clearly not maximal among proper ideals in $\mathbb Z$.

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I'm always amused by that joke, because I find the well-ordering theorem to be 'obviously true'. –  Hurkyl Dec 5 '12 at 19:33
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But then you'll also find those who think that the axiom of choice is "obviously false" :D –  Martin Argerami Dec 5 '12 at 19:47
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I disagree that Zorn's lemma gives no procedure for finding the maximal element. Start with an element, find a bigger one, find a bigger one, and so on until you can't get any bigger and you are done. (This procedure may require a transfinite recursion to finish) –  David Harris Dec 6 '12 at 0:09
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