Question: Can $\omega\_1$ (the first uncountable ordinal) be represented as union of an uncountable collection of cofinal, pairwise disjoint subsets?

Answer: 
For an explicit example of an uncountable partition of $\omega\_1$ into cofinal sets, for each $\alpha<\omega\_1$ let $A\_\alpha = \{ \beta<\omega\_1\mid \exists \gamma<\omega\_1,\thinspace \beta=\gamma +\omega^\alpha\}$. That is, partition $\omega\_1$ by looking at the last summand in the Cantor normal form of each element of $\omega\_1$.
Assuming the axiom of choice, the answer is yes. Under AC there is a bijection from $\omega\_1$ to $\omega\_1 \times \omega\_1$, so we can partition $\omega\_1$ into an uncountable collection of uncountable sets. Any uncountable subset of $\omega\_1$ is cofinal.

Edit: As Andres Caicedo points out in his comment (thanks!), AC is not needed.