Construct regular expression for language
Assuming that the input alphabet is $\Sigma = \{a,b\}$, the term $(a|b)^*a(a|b)^*b(a|b)^*$ describes all words where there is an $a$ followed by a $b$, and the term $(a|b)^*b(a|b)^*a(a|b)^*$ describes all words where there is a $b$ followed by an $a$. If we combine both, we get
$$
((a|b)^*a(a|b)^*b(a|b)^*)|((a|b)^*b(a|b)^*a(a|b)^*)
$$
which describes the set of all words in which at least one $a$ and one $b$ occurs.

Edit: An even shorter solution would be 
$$
(a|b)^*(ab|ba)(a|b)^*.
$$
However, this solution requires the alphabet to contain exactly $a$ and $b$ whereas the first solution can easily be extended to arbitrary alphabets.