Yes, the vagina does leak some quantity of goo for most of the time that it isn't leaking blood. No washing of the vagina itself, i.e. douching, is required for good hygiene (and can actually be counterproductive) but gunk and smell can build up unless the vulva is cleaned regularly. This has to be done with care as it's a sensitive area (obviously) and harsh soaps or perfumes can cause irritation. I shudder to contemplate life during parts of history when people did not bathe regularly.
Low-quality toilet paper is very unpleasant as women need to wipe between the legs after urinating (there's that sensitive vulvar skin again). At best the bad TP is just rough, at worst it can start coming apart and sticking in the labia. Yuck! Environment be damned, scratchy recycled toilet paper has no place in my household.
Wearing very tight pants, or anything that binds around the crotch, is uncomfortable as it squashes and pinches the vulva. Wearing pants without underwear to protect the vulva is also uncomfortable unless the pants are quite loose and soft. Chafing and wedgies are no fun.
Women have an advantage over men in that the most sensitive parts of our genitalia don't hang outside the body (see Dan Holliday's answer to What is it like to have a penis?) and are thus somewhat shielded from casual insult. However, we are more sensitive to intimate contact. If you will be interacting with a vagina and the area around it, please wash your hands, be careful about scratchy facial hair, and cut your fingernails. Also, lube is a wonderful invention, take advantage of it.
Our culture makes a really big deal about penetrative, penis-in-vagina sex. It certainly can feel good and have emotional significance depending on one's partner, but when just considering what maximizes sexual pleasure, for most women it's not that high on the list. The vagina itself has very few nerve endings compared with other areas (especially the clitoris) and many women find it difficult to orgasm from vaginal stimulation alone. Penetration together with clitoral stimulation can feel really great, though.
From a social and psychological standpoint:
The vagina is never neutral territory. Never. The vagina is always up for discussion, always the target of opinions and dictations. Opinions about whether too many people, or too few, have seen it, touched it, entered it. Opinions about whether those were the right kinds of people. Opinions about when you should give birth from it, and how often, and in what way. Opinions about whether you forfeit your right to agency over your vagina by dressing attractively, or getting drunk, or just being alone with a man. Opinions about whether merely having a vagina means that you forfeit your right to agency over seemingly unrelated matters. Opinions about whether the vagina is a sign of weakness or a seat of power, dark and primitive or lush and lifegiving, a mere receptacle for the spreading of seed and the notching of belts or a great treasure by which women tempt men and hold them in thrall.
With so many opinions flying around there is no right way to have a vagina. Seemingly neutral, personal issues take on social and political significance. Consider, for example, the seemingly trivial matter of grooming pubic hair. A woman who keeps her natural hair may be deemed unkempt and unfeminine. If she removes the hair she may be derided for bowing to patriarchal pressure, or even accused of encouraging men to prefer hairless, sexually immature lovers. Debates are fought on the field of women's bodies.
Having a vagina means grappling with a fundamental vulnerability. The vagina is the penetrated part, the pierced, the fucked. A woman who inclines towards being sexually aggressive or dominant must consider the tradeoff between experiencing the most rewarding sexual pleasure, and placing herself in a state that many view as inherently submissive, opening herself up to being touched in her most internal and intimate parts.
Adding to Jane Huang's answer (which I wholeheartedly agree with), and adding it as my own answer at her suggestion:
For more information, consider attending or otherwise viewing "The Vagina Monologues."
There's that whole, painful "having to shoot a huge baby out of a small hole," stretchy problem, too. So many books refer to the vagina being "elastic" in this case...but it's a bad comparison because elastic doesn't tear, and snaps completely back into shape after you stretch it. When your body changes from childbirth this way (and nobody warns you beforehand that it might, because they don't want to scare or discourage you) it is frustrating and can do a real number on your self-esteem. Some lucky women can afford vaginoplasty (plastic surgery) to correct any damage done by childbirth. It's not covered by insurance after the fact; some doctors will do what they can to repair you after you give birth, but not all of them will because they might view it as unnecessary.
Pertinent to the shape of our bodies, wedgies are probably 10x more painful for women than for our male counterparts. And one slang term for this particular body part is the "naughty purse." I'm here to tell you that most of us don't actually carry anything in our naughty purses except tampons during our periods. Really.
I won't even get into periods. They suck. Moving on...
Our examinations are a lot more invasive than examinations of male genitalia, which is both uncomfortable and embarrassing. This is because things can go wrong for us in places we're unable to see, whereas men can just look at themselves and know immediately whether all is well in Junkville. Stirrups, speculums, PAP smears...all are no fun because you have to have some stranger poking around "up in your business." (You guys just have to cough and take the occasional finger in the butt. We get probed and fisted.) Because of the concave, moist and wet nature of things, we get infections a lot more easily than men do, (it's actually very common with all women, and can happen no matter how clean your habits are) and when it happens it's an absolute drag (itching in a place you can't really scratch is no fun, and having to walk around with cream down there is pretty gross).
I'm sure there are cases where women are obsessed with penises (in general and specific ones), but I've known far more cases of men who are obsessed with what a specific woman's vagina looks like, what color her pubic hair is, wanting to "be inside of it." This is pretty annoying because most of us don't understand the big deal. I honestly believe that the frustration some people describe in women as "penis envy" has nothing whatsoever to do with penises and everything to do with what a pain our own equipment is.
We feel like we're expected to wax, or trim, or shave little designs down there for you guys...or even vajazzle (http://www.vajazzling.com/)...we don't do that stuff for ourselves, I promise you. If the world had no men in it, we'd all just let it grow. ;)
And, speaking of groups of women. I know a lot of men think that when women get together, we talk about our vaginas and our periods. This is largely untrue unless there is a medical problem with one of those things that we want advice on. (Though, I have never once experienced running with a female friend along a beach, only to have her ask out of nowhere, "Ever have those times where you just don't feel fresh?" which happens in TV commercials and magazine ads for feminine products. That does not happen.) Actually, we talk about what's going on in our lives, books, movies, recipes our kids, where to buy cute shoes...and how hilarious it is that guys think we're showering together, having topless pillow fights in slow motion, or talking about our vajayjays.
It seems like it should be a cavern like a nose nostril, or the opening of your ear - yet it is not. So when there is something inside of it like a tampon, finger or penis or baby head, the inner folds move and expand yet stay tight against the item so that no air is allowed to enter the body.
The only way for a doctor to see inside is to insert a speculum and spread it far apart so he can peak inside and see the cervix and the inner walls.
The vagina's claim to fame is that it completely encompasses whatever enters it, it can tighten around the item with the use of muscles, it can warm the item with the use of excess blood and lubrication and it remains so until you are done with it.
So what if it is yours? Well, just like all body parts, you are aware of it just by thinking about it. If you have to go to the bathroom you squeeze those same muscles to stop the bladder flow. However the bladder is not inside of the vagina. It is outside of the vagina yet the urine seems to come from the inside of the vagina but it does not. It comes from a tiny secret opening right above the vagina.
A lot of people think that if you urinate after sex that the urine will kill the sperm. Not so. The urine never enters the vagina which is where the sperm is. By the time the sperm has entered the vagina it is too late for anything to kill it. Even if you spray water from a shower nozzle up there you can't be assured of getting all of it. Those little swimmers are fast. Once they enter the cervix they are long gone.
Now, there is something called a yeast infection. Sounds gross, looks gross, but how does it feel? Like hundreds of ants crawling inside and out. The only way to kill the ants is to squirt them with cream, but they don't die right away. First they mount an attack against the cream and the battle is worse than the infection. After they are all finally dead, and you swear that you will never take antibiotics again, there is this thing called vagina memory loss. You can't possibly believe how bad it was so you do it again. Same with childbirth. Oh, it couldn't have been that bad!
You would think that gravity would have everything spilling out of the vagina, onto the underwear, down the legs, but with those muscles, clenching the opening shut, not much escapes quickly with the exception of amniotic fluid when the bag breaks during childbirth. The rest of the things that escape do so slowly and covertly. Generally because there are not many nerve endings surrounding the vagina, you can not tell when something is coming out from within the vagina itself.
So what is it like? Well, it's there, doing its job. 10 partners, 6 pregnancies, 2 episiotomies, 3 abortions, a hysterectomy and ~600 orgasms later, it's still there, ready for action.
Edit: I went for a humorous cheap shot, mostly because men seem to think women are super mysterious and hard to understand. I can't believe guys get so worked up about not having a pet vagina in their lives. Vaginas are poorly designed pieces of equipment that leak goo, and sometimes blood, and even more rarely, small children. The end.
I agree that seeing The Vagina Monologues is an important step in understanding the women of the world. And Betty White recently wondered out loud: "Why do we always think that men have the tough parts? Those balls are tender and sensitive. Now a vagina, that can take a pounding!" Seriously, the vagina or Yoni in Sanskrit, is the most wonderous Jade Garden of them all. It is a direct lineage to Transendence and Enlightenment if used in wise and resourceful ways. Contrary to popular belief there are several nerves that supply the vaginal recesses and these nerves can deliver far more satisfying orgasms than the clitoris could ever be credited with. If we were a culture that celebrated the blood rites and revered women we wouldn't be worrying about 2012 right now. Or climate change. Or our political situation. Or rape and horror stories about infibulation and other attrocities that are performed on women for men.