I keep seeing "third wave feminism" being used, usually in a derogatory way. Is it an arbitrary distinction, or can feminist movements really be divided in three categories?
The wave paradigm is definitely one of the ways historians and other scholars differentiate between various generations of feminists.The waves analogy is used because it not only distinguished between different generational cohorts, but also the differing agendas and goals of feminism. The general schema is that first-wave feminism was mostly preoccupied with the issues of suffrage and various other related issues; the second-wave was largely a postwar phenomena with roots stretching back to the 1920s and dealt with wider civil rights and the meaning of gender; the third-wave largely grew out of the ideas of the second and incorporated different approaches to gender such as race or socioeconomic class. There is considerable debate as to whether or not in 2017 there is a fourth-wave of feminism, which highlights some of the dangers of reducing things into a chronological schema.
Obviously, this paradigm does have some flaws. For one thing, it is a very American-centric approach to the history of feminism and defines what was often a global movement with institutional and personal linkages with developments in the US. While American feminist thinkers have been very influential within feminist thought, they are not operating in a vacuum and an American-focus often sidelines or ignores the developments in feminism in non-Western countries. Additionally, the transition points between waves is often rather messy. For example, there is a huge gap in the wave timeline between the passage of the 19th Amendment in 1920 and the publication of The Feminine Mystique in 1963; both events are landmarks for their respective waves, buu it was not as if feminists stopped what they were doing between 1921-1962. The wave paradigm also obscures some of the divisions and developments within each wave. There was a clear generational divide between veteran suffragettes like Carrie Chapman Catt, who favored working with existing channels to gain suffrage, and younger suffragettes like Lucy Burns and Alice Paul that championed more strident civil disobedience and a more direct protest for the right to vote. The goal may have been the same, but the tactics were different and this reflected some of the generational differences between women. Older suffragettes in the 1900s were often mentored by abolitionists and cut their political teeth in the temperance movement while the younger generations were byproducts of growing elite institutions of the Gilded Age such as the Seven Sisters. Wave models often obscure the cleavages within these groups such as race or sexual orientation. The US suffrage movement in the 1900s counted among its ranks not only Ida B. Wells, a noted African-American journalist and muckraker, but also Rebecca Felton, a Georgian woman who justified lynching and was the last member of Congress to have owned slaves.
One common thread between waves though is a degree of denigration of feminism by wider society. Such denigration is often sexual in nature, skewing to extremes of mainstream sexual desirability and posits that the feminism is a subversion of the natural domestic order. This cartoon from the early twentieth century shows both extremes in which a suffragette starts off as an immature coquette, and then becomes gradually deranged from the lack of a marriage in her life. This 1971 cartoon features "women's lib" activists not only as harpies, but also as failed mothers as one younger daughter saying "Mommy, I got a boy in trouble," with the implication she's pregnant by virtue of the bulge-hiding trench coat. Another 1970 cartoon from the same artist shows the inversion of domestic responsibilities and motherly abandonment. This cartoon takes on the stereotype of bra-burning where a small-chested feminist puts on a fake set of breasts that jiggle before going out with her placard, implying that feminists are using their breasts garner attention. One of the more extreme views of the sexless feminist 1995 cartoon shows a feminist as an angry cross between a Valkyrie and a neanderthal irrationally demanding that nighttime be free from the dangers of sexual assault (she wears a "survivor" armband, but the cartoon's visual language implies this is false- who would assault a 300 lbs. angry she-viking?). Such stereotypes and attacks are quite common against feminism in various corners of the internet and rely upon much the same visual typologies. Feminist women are simultaneously sluttish and unf**ckable, conniving and naive, pathetic and an existential threat to the natural order. This is but a very small sample of a larger process of denigration that dismisses the ideas feminists bring up by directly connecting them to women's bodies and issues of sexual desirability by men or domesticity.
The wave paradigm is still a useful analogy despite its shortcomings. The metaphor actually works rather well if one considers the feminist movement as cresting and crashing as it achieves certain goals like suffrage, but the wider aim of a more equitable society has proven quite elusive. Gender historians and Americanists (again, the wave model does not fit well outside of the Anglosphere) use the wave as a means to describe and differentiate between different feminisms over time, and usually with the caveat that the feminists in these waves did not define themselves in such terms.