A Bigender person is a person who feels that their gender is fully male and fully female, or any pair of genders, generally by switching between the two.  For some people, the switch is between two very distinct genders, which some consider to be separate "facets" or "personas".  For others, the switch is much more fluid, with more grey area.  

Transitioning is a difficult idea for many bigenders.  Because they have both a male and female gender, assuming they aren't intersexed, then when they're in the gender that matches their body they're as happy with their body as a Cisgender person, but when they're in the gender that doesn't match their body they feel the same dysphoria and need to transition that most Transgender people feel.  It's not currently possible to create a body that changes from male to female at will, so it can be difficult for a bigender person to achieve a presentation or body that allows them to be comfortable in both genders.

The difference from an androgyne is in the integration of the two genders: androgynes are one gender which incorporates feminine with masculine characteristics into one more-or-less coherent whole, while bigenders are two genders that are expressed separately. There's also a difference of degree: androgynes are neither fully man nor fully woman, while bigenders can switch between identifying fully as one gender and fully as another.

As with all people, both genders in a bigendered person can be masculine, feminine, or androgynous.