The statement of Zorn's lemma in the book "Modern methods of mathematical physics: Vol $1$" by Simon and Reed reads as follows:
Let $X$ be a nonempty partially ordered set with the property that every linearly ordered subset has an upper bound in $X$. Then each linearly ordered set has some upper bound that is also a maximal element of $X$.
I am confused about what does the second sentence mean. Does it mean that every linearly ordered subset of $X$ has a common upper bound which is the maximal element?
No. It means that if every linearly ordered subset [read: chain] has an upper bound, it does not have to be within that chain (although it could be if the chain has a largest point), then there is a maximal element in the entire order.
Recall that maximal simply means that there is no strictly larger elements in the order, it does not mean largest.
For example, if you consider $V$ to be a vector space over some field, say $\Bbb R$, then we can look at the partial order of all linearly independent subsets of $V$. We can verify that given a chain of linearly independent sets, its union is also linearly independent. It does not necessarily mean that the chain has a largest element, but it could be. Zorn's Lemma says that this partial order has a maximal element, and we can prove that any such maximal element is a basis for $V$. But we can easily replace one basis element by a different one (e.g. if $v$ is in the basis, we can replace it by $2v$), so you can see that this maximal is very much non-unique.