Gender identity relates to one's feeling of gender within themselves. The most common gender identity is "cisgendered", an identity so common most people don't need to know it. Gender identity may or may not vary from physical body and typical gender roles/expressions.
NOTE: I see labels as a rallying point to find people like yourself, not as a box to limit you. Positive uses for labels include exploring ideas and themes around your identity, knowing what words to google and finding people to talk to.
Cisgender indicates that someone identifies with the gender that they were designated at birth. It is the opposite of "transgender" and the most common gender identity.
Transgender people are people who have a gender identity, or gender expression, that differs from their sex assigned at birth.
As an umbrella term, transgender includes bigender, gender-fluid and other non-cis gender identities.
Many transgender people are basically binary-gendered, meaning they feel they are "male" or "female" and just want their body and lives to match that polarised gender type.
Bigender identity that includes any two gender identities and behaviors, possibly depending on context. Some bigender individuals express two distinct "female" and "male" personas, feminine and masculine respectively; others find that they identify as two genders simultaneously.
Genderqueer is a catch-all category for gender identities that are not exclusively masculine or feminine—identities which are thus outside of the gender binary.
Genderqueer people may have an overlap of gender identities, view themselves as having two or more genders, no gender, or fluctuating or fluid gender or just don't have a recognised name for their gender.