(Note: This was intended to be a comment to unknown (google)'s answer - but as I'm new here I can't post comments.)

As Pete L. Clark points out, unknown (google)'s answer is false as stated.
However, this is only because of the omission of the word "infinite".
A correct statement is:

An infinite profinite group $G$ is homeomorphic to $\{0,1\}^{w(G)}$,
where $\{0,1\}$ is the $2$-point discrete space,
and $w(G)$ is the weight of $G$.

This is Theorem 9.15 (pages 95-98) of "Abstract Harmonic Analysis I" by Edwin Hewitt and Kenneth A. Ross. (Hewitt and Ross actually state the result using the minimum cardinality of a local base at $1_G$, rather than the weight of $G$, but these are equal for infinite profinite groups.)

Notice that the case of countable weight is an immediate consequence of the usual characterisation of the Cantor set.