It is alleged, with rather alarming frequency, that Tolkien revealed himself through his writings to be misogynistic: that he has no strong female characters, and that all his women are down-trodden, servile, and in general, stereotypes from that mythical 'golden age' when women stayed in the kitchen while men did the interesting stuff. My reaction to this is: have the people who say it actually read Tolkien?
It is true that there are not, comparatively, very many women in Tolkien's writings - just as there weren't in his life as an Oxford scholar. It may be that this is symptomatic of the times Tolkien lived in - but to my mind, it's far more typical of the man himself. It's not that he considered women inferior - it's that, in his insulated world, he usually didn't think about them at all.
But an argument from absence isn't much of an argument. To find out what Tolkien thought of women, we need to look at the ones he did include in his writings. Unexpectedly, at least to anyone who accepts the statement that Tolkien was a misogynist, they are almost all 'strong' characters - and in many different ways (not just 'better at fighting', which is what many people seem to think):
The Women of the Third Age
Eowyn: Probably the woman with the most description, Eowyn actually is an oppressed woman, probably the only one Tolkien gives us. She is expected to look after the household while the men of Rohan do the fighting. And how is this portrayed? Very negatively. As she puts it: "All your words are but to say: you are a woman, and your part is in the house. But when the men have died in battle and honour, you have leave to be burned in the house, for the men will need it no more."
And Eowyn wins. Through her own skill and cunning, she joins the Muster of Rohan (and unlike in the film, manages to keep this secret from everyone); through her strength of both arm and will, she kills the Lord of the Nazgul. Then, yes, she falls in love... but there's no indication that she became a meek, domestic wife. She and Faramir go off to rule Ithilien - a decayed garden realm where orcs undoubtedly still sneak out of the mountains at night. She becomes a healer - but I doubt she threw away her sword.
Galadriel: The clear ruler of Lorien, Galadriel is possibly the most powerful non-demigod in Third Age Middle-earth. In her appearance in the narrative, we find her warding off Sauron's mental attacks on a constant basis, sustaining an impossible land (it's clear that both the temporal distortion the Fellowship experience, and the very fact that mellyrn grow in Lorien, are due to Galadriel and her ring), and acting as both a guide and mentor to Frodo. Offscreen, she even exhibits military strength - it is Galadriel, by her own innate power, who tears down the fortress of Dol Guldur, after fighting Sauron's armies all the way from Lorien.
Arwen: Arwen is the character most people are thinking of when they say Tolkien's women are oppressed. And yes, her sole role in the book is to sew a banner and then get married. Oh, and to give up the prospect of immortality in order to pursue her desires. Is it weak to sacrifice for something you want? Maybe, if you're forced into it - but I don't see how Arwen could have been. I don't see Elrond pushing her into marrying Aragorn, and while Galadriel (her grandmother) clearly supports it, I can't picture her as forcing a marriage, either. Literally all Arwen had to do was stay in Lothlorien for a bit longer - a mere hundred years, say. She had already been there a long time when Aragorn met her there. Arwen may not have done much, but her strength is in her sacrifice - in giving up so much to secure the future of Middle-earth, and her chosen husband. No, she's not a fighter - but not everyone has to be.
Lobelia: Lobelia Sackville-Baggins is a minor villain, but one with extraordinary persistence. Her husband (what was his name again?) is irrelevant: it is she who wants Bag-End, and waits over fifty years to get it. Even when she is thrown in prison, she endures - she comes out as irascible as ever, despite everything that's thrown at her. I don't like the woman, but she is definitely a strong character.
Assorted Wives: The various wives and mothers barely, if ever, appear on screen - Rose Cotton shows up very briefly, while Dis (sister of Thorin), Estella Bolger, Diamond of Long Cleave, and all the others are essentially names. For the most part, they seem to constitute rewards for our heroes, but it isn't quite as simple as it looks: there's a whole other Theory about this.
The Women of the Second Age
Galadriel: Yes, again. In the Second Age, Galadriel (with Celeborn) founded the elven realm of Eregion/Hollin - the place where the Great Rings were eventually forged. She later left, though, moving to what would later be known as Lothlorien - the forest that, in the Third Age, she would come to rule. It's also worth noting that of the Three Elven Rings, Celebrimbor originally gave two to Gil-Galad, the High King. The last, he passed to Galadriel.
Erendis: Oh, Erendis. One of the great tragedies is that Tolkien never finished any of his Numenor stories - particularly the tale of Aldarion and Erendis. Aldarion, crown prince of Numenor, marries the lovely Erendis - but repeatedly abandons her in favour of his first love, sea voyages. Eventually, Erendis actually leaves him - an event that only occurs three times in Tolkien's works (the other two being Aredhel and Nerdanel). That takes a strength that even many people nowadays lack - the will to get out of a relationship gone wrong. It should also be noted that Erendis doesn't just spend the story moping prior to that point, either -she does want their marriage to work, and does everything in her power to get Aldarion to see that.
Tar-Miriel: And the prize for 'most oppressed woman in the history of Middle-earth' goes to...! Poor Tar-Miriel (out of respect for who I will always call by her rightful title as Ruling Queen of Numenor). Her despicable cousin Pharazon (again, I refuse to use the 'Ar-' title of this usurper king) forced her into an illegal marriage (they were first cousins, making it against the laws of Numenor) in order to steal her country (and yes, Tolkien pretty clearly implies he raped her, too), then proceeded to turn Numenor against everything she held dear. He brought the Dark Lord Sauron to the island, for Pity's sake!
So where does her strength come in? Tar-Miriel, after all, seems to have done nothing during Pharazon's rule - nothing to turn him away from his dark path. Of course not - that isn't her strength. Her strength lies in the same quiet endurance as Elrond in Rivendell. Despite everything that happens, Miriel keeps her faith - and, at the very end of her life, when her only allies (Elendil and company) have fled, and she is alone, the last of the Faithful in Numenor - even as the waves come she tries to climb the Holy Mountain, to appeal to Iluvatar at the place where only she is permitted to, and the only place she is permitted to. Can we believe she was doing this to try to save her own life? Hardly, or she would have left with Elendil. No - at the very last, Tar-Miriel, true twenty-fifth ruler of Numenor, tried to intercede on behalf of her people, and her country.
The Women of the First Age
Galadriel: Yet again. When Feanor rebelled against the Valar, the lords of the Noldor split into two groups: those who agreed with Feanor that they needed to pursue Morgoth, and those who thought they should stay in Tirion. The only exception to this, that we know of, is Galadriel: she wanted to leave Valinor, but not for Feanor's reasons. Where everyone else was caught up in the moment, Galadriel was planning ahead, already exploring new lands in her mind.
And that's not the only instance where she's exception. During the Kinslaying at Alqualonde, we're told that Fingon joined Feanor in the battle, thinking that the Teleri had attacked him (when in fact, it was the other way round). Galadriel, apparently alone amongst the Noldor, instead defended her Telerin kin, fighting fiercely on their part against the Noldor.
Luthien: Luthien Tinuviel is probably the most fleshed-out female character in The Silmarillion - so much so that I have an entire other page about her.
Melian: Melian is one of the few Maiar to actually choose her own path in life - she left Valinor to explore the world, and when she found something she liked, she actually stayed there. She is also one of the most magically powerful characters in Middle-earth; the Girdle of Melian was strong enough to hold every evil force out of Doriath for hundreds of years. And, later, when her daughter Luthien wanted to go after Beren, Melian did one of the strongest things in the entire legendarium: in spite of all her godly powers and foresight, and the knowledge of what could happen to her daughter, she let her escape.
Haleth: The Lady Haleth is one of Tolkien's few unmarried women - because she was too busy fighting off the orcs that killed her family, taking over the rulership of her people, refusing the advances of Caranthir son of Feanor (always a good idea), leading the Haladin through a giant-spider-infested valley to find a new home, telling Thingol (the most powerful king in Middle-earth) that Brethil was hers now, and generally being the strongest ruler of the Edain described in the Silmarillion. I dare you to tell her she's not a strong female character. Did I mention she has an axe? Her early life is discussed further here.
Miriel: Miriel Serinde is probably the physically weakest elven woman we can read about - she actually dies of exhaustion from giving birth to Feanor. But her character really develops after that - in the time when she is in the Halls of Mandos, listening to her husband trying to persuade her to think of him over herself, and return to life. In defiance of Finwe, Feanor, Mandos, and Manwe, Miriel chooses her own path: to stay in the Halls forever, the first of the Eldar in Aman to taste death.
Aredhel: Aredhel is a woman used to getting her own way. She leaves Gondolin in defiance of her brother the king, only to be captured by Eol the Dark Elf. While not precisely forced into marriage, she didn't exactly go freely, either - and so, when she got a chance, she left again, forsaking her husband despite his 'strength'. I don't particularly like her, but you can't deny she had strength of character and will.
Elwing: Elwing is one of those characters who spends most of her time being dragged around by other people; nevertheless, after her husband Earendil goes sailing off, and the Sons of Feanor attack, she is willing to sacrifice everything - her sons, and even her life - to keep them from gaining the Silmaril. And later, when their ship reaches the Undying Lands, Earendil tells her to stay on board for her own safety - to which Elwing says, in essence, 'To Mandos with that - if you're going to face death, I'm facing it right alongside you.'
All the rest: There are far more women in The Silmarillion and its associated works than in The Lord of the Rings, and I could write a paragraph like the above for most of them. Morwen, Finduilas, Idril, Nienor, Nerdanel, even Powers like Yavanna or Uinen - in their different ways, they are all strong characters.
If I had to choose one word to characterise Tolkien's women, it wouldn't be 'weakness', or 'oppression' - it would be 'refusal'. Every woman who actually has a character refuses to accept the world as it is presented to her, whether it is Eowyn's refusal of the role her uncle chose, Haleth's refusal to surrender her leadership, or Miriel's refusal to bow to her husband's pressure and return to life. Some of them protected their opinions with sword or spear, some with magic, and some with sheer stubborn willpower - but they all refused, and they are all strong.