Sidhe is just the older spelling. The reform was in the 1940s or 1950s, so you still see a fair bit of the old system around, written in the cló gaelach, e.g. on street signs and monuments in Ireland.
Goidelic is the name of a the language family (a subgroup of Celtic) that contains Irish (Gaeilge), Scots Gaelic (Gàidhlig), and Manx (Gaelg).
The Irish word for "woman" is bean. But feminine nouns are lenited after the singular definite article an, and bean is a feminine noun, so you say an bhean, not an bean.
There are other things that can cause lenition, and there are times when bean will be eclipsed (mbean), but for now, when you've encountered bhean, it has probably been an bhean.