A Bigender person is a person who feels that their identity 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 bigender people. Because they often have both a male (or masculine) and female (or feminine) gender, many bigender people may feel one identity is in as much harmony with their body as a cis gender person, but their other identity may make them feel the same dysphoria and/or need to transition that many binary 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 many bigender people 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 bigender person can be masculine, feminine, or androgynous.