Gender
The word gender, originating from the Latin genus (sort, kind) has gained several meanings over the centuries and especially during the past decades. Significant debate exists between ideological groups that wish to highlight the importance of one particular definition of the word.
The words feminine and masculine are usually associated with concepts of gender, as opposed to the words female and male which more commonly relate to sex, though all of these words and concepts are frequently conflated by the general public.
Original definitions
Grammatical gender
Older and mostly non-political definitions of gender include grammatical gender, which is merely the division of certain words in some languages among two or more categories to highlight a (presumed) gender of the object, or lack thereof. This gendering of words is sometimes applied rather nonsensically, such as in German and French, where all sorts of inanimate objects from furniture all the way to planets are accorded a gender, whereas some truly sex-specific words can surprisingly have a neutral gender, such as mädchen, the German word for "girl", which is accorded a neutral gender despite the words "woman", "boy" and "man" being gendered in accordance with the human sex they correspond to. References to grammatical gender seem to date back to the 14th century.
The words feminine and masculine are often used to refer to grammatical genders, although terms such as female pronouns and male pronouns are also commonly used to refer to grammatical constructs.
Synonym to sex
Another old and mostly non-political definition of gender is as a synonym to sex, i.e. the categorization of an organism with respect to its role in binary reproduction. This use of the word is popular presumably because of the vulgar connotations the word "sex" might be perceived to have. The use of "gender" as a synonym for "sex" likewise seem to date back all the way to the 15th century, and can be found even in academic texts which clearly deal with a strictly biology-based reproductive categorization of organisms.
Similar to the use of gender to mean sex, the words "feminine" and "masculine" are also sometimes used as close approximations of "female" and "male". In biology, it is not uncommon to hear terms such as "feminization" or "masculinization" when referring to anatomic developments associated with the female and male sexes. Similarly, the euphemism feminine hygiene products is often used to refer to menstrual hygiene products, relating clearly to the female sex.
Recent changes
In the 1950s, psychologist John Money began introducing alternative definitions of gender in his texts, in particular terms such as gender role and gender identity. This might be the origin of the understanding of gender as a social construct, although Money's theories are not necessarily compatible with contemporary interpretations of this concept.
What is meant with gender as a social construct is the realization that words such as "woman" and "man" (also "girl" and "boy") do not merely induce thoughts of just any human that happens to be anatomically female or male. Rather, on hearing these words (or their translation to one's native language), one is likely to imagine a person who not only belongs to a certain anatomic sex category, but also fulfills certain expectations with regard to clothing, speech, behavior, and so on. For instance, upon hearing "woman", one is more likely to think of a person with long hair and perhaps a dress, even though both of these "gender markers" have no relation to female anatomy. (Male humans can also grow long hair, and wear dresses.) These gender markers/expectations differ from culture to culture and across history, and include many aspects with no relation to sexual anatomy. As such, they are understood to constitute a social construct of gender, that is separate from the anatomical facts of being female or male.
Onwards from the point of recognizing gender as a social construct, at least two competing theories regarding the deeper nature of gender have evolved: gender as a tool of sex-based oppression, and gender as an innate identity.
In relation to these developments, the words "feminine" and "masculine" have likewise taken on meanings that more strongly emphasize the stereotyping aspect of the social construct of gender. Femininity might refer to a personal style that includes long hair, dresses, the color pink, softer expressions, etc., whereas masculinity might refer to an opposite personal style.
Gender as a tool of oppression
According to feminist theory, in particular second-wave radical feminism, the social construct of gender is more or less synonymous with sex-based roles and stereotypes imposed forcefully on women, which serve to keep intact their inferior position in a male-supremacist society. The personality traits associated with women are believed to be outright myths, or self-fulfilling prophecies that are established through the years-long socialization of female children into the role of a "proper girl" and later a woman. These personality traits ascribed to women and girls might include some positive aspects such as grace and kindness, but combined with the expectation of passivity and docileness they easily lead to a personality that makes it easy for men to exploit women's labor and bodies.
The feminist realization of gender as a social construct actually dates back to before John Money's use of the word in his works. In her 1949 book The Second Sex, Simone de Beauvoir famously spoke of how one is not born, but becomes a woman. (Volume II, Part One, Chapter 1) In the book, she elaborates on the ways in which societal forces lead persons born to the female sex to become "women" in the way society expects it. Although de Beauvoir might not have explicitly mentioned a "separation between sex and gender" as is commonly heard in feminist circles nowadays, or otherwise used the word "gender" in the way described here, her analysis nevertheless seems to cover the same concept.
Andrea Dworkin, in her book Our Blood, writes the following about the "system of gender polarity":
“... I have made this distinction between truth and reality in order to enable me to say something very simple: that while the system of gender polarity is real, it is not true. It is not true that there are two sexes which are discrete and opposite, which are polar, which unite naturally and self-evidently into a harmonious whole. It is not true that the male embodies both positive and neutral human qualities and potentialities in contrast to the female who is female, according to Aristotle and all of male culture, “by virtue of a certain lack of qualities.” And once we do not accept the notion that men are positive and women are negative, we are essentially rejecting the notion that there are men and women at all. In other words, the system based on this polar model of existence is absolutely real; but the model itself is not true. We are living imprisoned inside a pernicious delusion, a delusion on which all reality as we know it is predicated.
In my view, those of us who are women inside this system of reality will never be free until the delusion of sexual polarity is destroyed and until the system of reality based on it is eradicated entirely from human society and from human memory. This is the notion of cultural transformation at the heart of feminism. This is the revolutionary possibility inherent in the feminist struggle. ...”
A naive reading of both de Beauvoir and Dworkin can lead to a confusion where one is led to believe that their writings are in line with contemporary "gender identity" theory. Dworkin for instance writes that "it is not true that there are two sexes which are discrete and opposite". (It's possible to give multiple interpretations to the clause "... which are discrete and opposite." Was she denying binary reproductive sex?) However, both authors make extensive mention of female anatomy in relation to experiences of female oppression. For instance, in the very same chapter from which the aforementioned quote is taken, Dworkin speaks of IUDs for female birth control, and of the clitoris as the source of female sexual pleasure. As such, the compatibility of her theory with contemporary gender identity theory is rather questionable.
Gender as innate identity
(This section needs elaboration.)
Most transgender activists use gender to refer to a supposedly essential, inherent identity of human beings, which determines whether they are a "woman" or a "man" (or something else) without any relation to their reproductive capabilities.
Theories around this notion are often associated with queer theory which in turn is based on post-structuralist philosophy. Writings based on these ideologies tend to be rather difficult to understand (sometimes arguably in an attempt of the authors to sound more sophisticated than they are), and often have no completely clear conclusions. This might be why even among transgender activists, there seems to be no consensus on some questions one might think are of central importance, such as: is gender identity inborn and immutable, or is it a personal choice? If it's immutable, why do some people change their minds several times, and how can it be tested for objectively? If there are any objective measures of gender identity, what are they, if not an identification with sexist stereotypes of femininity or masculinity? For instance, is gender dysphoria a necessary condition for being considered "truly" transgender? If it's rather based on purely subjective notions, how can it have an effect on objective, material systems of oppression that affect people starting from birth? If deciding that one has a "female gender identity" means that one has really always been a girl/woman, does that mean one never benefitted from male privilege, even after living 40+ years as a man? If a "female gender identity" already makes a person female, does that mean bodily changes are truly unnecessary? Does this mean that a tall, broad-shouldered person with a full beard, a deep voice, coarse body hair, and intact male genitals can nevertheless be considered a female person? Does this person then have a right to access female-only facilities and services without restriction?
The unwillingness or inability of transgender activists to answer such questions (or alternatively, the absurd answers they provide in attempts to be consistent with their own core ideology) has led many feminists to feel disillusioned with the transgender movement and begin to see it as being incompatible with feminism. The observation that women who begin to raise critical voices are often met with overt hostility and shunning has further worsened the situation. By now, many feminist groups and organizations have begun to see "gender identity" as a misogynist belief system, which should be openly challenged.