In discussions about feminism, the sex of a person is a biological characteristic and their gender is a social role.
In biological, medical or mainstream discussions this distinction is not as commonly understood (or accepted) but distinguishing the terms in this way is important in feminist discussions. Gender is more commonly discussed than sex in feminist spaces.
Sex
Human sexes are distinguished in a number of ways, most commonly genitalia at birth (or in Western society, increasingly before birth due to ultrasound imaging), but also medical analysis of hormone levels and genetic material. The sex of a human is not a binary: quite a large number of people have some biological traits that are male and some that are female, or in some cases a variant that is not assigned a side of the binary. Intersex people who cannot easily be assigned a sex at birth due to having ambiguous sex characteristics have widely been discriminated against, and used to be (and often still are) 'corrected' surgically as infants to appear to be more typical of one sex due to the preferences of medical staff and/or their own families, often with devastating medical, emotional and social consequences later in their lives.
As an example of non-binary sex determinations, the 'typical' male sex chromosomes are XY and female XX. But there are XY people who exhibit almost entirely female sex characteristics due to, for example, androgen insensitivity and there are many people who have sex chromosomes that are neither XX nor XY.
Gender
A person's gender is a social role. Very many things play into gender presentation, including but not limited to: dress and appearance choices, choice of hobbies, choice of career, social behaviour and sexual behaviour. This social presentation is called "performance".
Gender being a social role does not imply that it is freely chosen or fluid for most people. Gender is also assigned to almost everyone at or before birth based on the determination of their sex. Most people consider their gender a very central part of their identity and the extent to which at least this core identification, if not every aspect of gender, is a biological property (of brain structure, or similar) is debated. The ability to vary from assigned gender roles is strictly policed in human society.
People who identify as the 'other' side of the gender binary from their biological sex use the term transgender (other people are cisgender). People who do not identify exclusively or primarily with either side of the binary use the term genderqueer.