The TL;DR Version
Do you have to wear white on Yom Kippur? No. Will you be weird (or the only one) if you don't wear white to synagogue on Yom Kippur? Nope.
The RTFM Version
Wearing white (presumably only to synagogue because what else are you doing while fasting?) on Yom Kippur is a very common custom, yet it's not universally done. As I understand it, it's a custom, not in the "custom that has becoming as binding as halacha" legalistic sense; instead, it's a custom in the "that's nice, what a meaningful and/or cute idea" sense. Symbolically, it's considered a reminder of purity and being "white as snow" after repenting (making teshuva). Apparently it also has something to do with angels, but I wonder how influenced the "angels-wear-white" idea is by Renaissance art. Personally, that's not a symbolism that resonates with me, and that's ok if some of the symbolism doesn't do it for you either.
You can learn more about the various symbolic meanings of white and other Yom Kippur customs at Chabad, the JewFAQ, and Aish. Unfortunately, I didn't find anything about the origins of this custom and when it became encoded or common. It's an easy logical jump to make because of the very common symbolism, so it's possible that it developed simultaneously (or nearly so) in different communities without normal cultural diffusion spreading it initially.
The amount of white being worn by the people in synagogue will differ from community to community and even year to year. One year in one town, you might be the only person not wearing white. And the next year in another town, you might be the only one wearing white. Most often, there will be some wearing white and some not. As far as I can tell, there is no stigma attached to not wearing white. Some people do, some don't, some forgot (uhh...been guilty of that more than once), some didn't get the laundry done in time, and some can't afford a white outfit they'd only wear once a year.
It's often associated with orthodox congregations but has been steadily growing in many reform and conservative communities (and may even be the community practice in some shuls). As a dimestore psychologist, I think there are two major factors pushing the increase in liberal communities: 1) the internet has made it easier to know about these customs from other communities, leading to a "that's cool, let's do that" change. A lot of people see deep meaning in this custom, and some believe it's beautiful. 2) A reaction against the perception of materialism that has plagued liberal High Holiday services since the early 1900s. I've spoken to some people who see wearing simple white clothing as a rejection of the "wear your best dress and fur to shul because everyone will be there" behavior and mentality that used to be more commonplace in many communities. Some shuls are still like that, but very few are defined by it today, in my experience.
So if you own some white clothes and they're clean, wear 'em. Or don't. Whatevs. If you forget, don't worry about it.
The kittle happens to be white and is often worn on Yom Kippur...but not because of this custom. It's a separate custom and a separate analysis.
We don't wear white on Yom Kippur because men wear kittles on Yom Kippur and kittles happen to be white. They're two distinct customs that may or may not overlap in your particular community. Maybe nearly every married man wears a kittle yet no one else wears white, or maybe there's only one or two kittles and the community overwhelmingly is dressed in their best whites. Or both. Or neither. But in an orthodox community, you won't find any women wearing kittles. Personally, I've never seen a woman (or even many men) wearing a kittle in the liberal congregations I've belonged to or visited. Your mileage will vary.
The kittle is the white, plain garment that is worn when one is buried. Yep, morbid. Interestingly, kittles made for living people to wear are often more elaborate and may include pockets. Yes, there's kittles for live people and ones for dead people. We're so weird. Moving on.
Men tend to wear the kittle before death, primarily as a reminder of mortality, though I think most men associate it more with Pesach and Yom Kippur than coffins. Funny how things get backward, right? People do the funniest mind tricks to forget that we're all mortal and will RETURN TO THE DUST. /drama
Is it custom or halacha? Resident Male Expert, my husband, says it's a custom. He wears a kittle, so I figure he knows something about them. Given the unpredictability of kittle-wearing in the modern orthodox community, I believe it likely is "just" a custom, though a common one. In some communities, particularly chareidi communities, there may be tremendous peer pressure (perhaps even disguised as "halacha") to wear a kittle at certain times because "that's just what's done." Know your community and avoid unpleasant surprises.
The kittle is generally first worn at one's wedding, as well as at each Pesach seder and Yom Kippur after. If one does that. Not everyone does. Some don't wear it at the wedding but wear it some chagim. Some wear it at the wedding and never wear it again until aforementioned coffin. Each person, and even each year, is different. And maybe you forgot to get your kittle drycleaned...or to pick it up from the drycleaner. No big deal according to most interpretations of halacha and custom. However, your mileage may vary, as mentioned above.
Tips & Tricks
So how does one "wear white"? You'd think that's an easy question, right? People wear white in many different ways.
First off, you will probably see a healthy showing of off-whites because apparently cream and beige count as white. I get it, and I've even done it, because owning white clothing is usually an impractical idea. Especially white pants and skirts. Most people don't have the spare income to buy white clothing for one day a year when it's not actually required, not even by peer pressure.
Second, many people wear only some white clothing, not an entire white outfit. Usually a shirt because, again, white pants and skirts are basically a terrible idea any day of the year. Also, they're harder to find in most fashion seasons. Married women who cover their hair and dress is white on Yom Kippur often wear a white scarf or hat. Again, not required in any sense of the word, especially if you normally cover your hair with a wig. Aish has an "Ask the Rabbi" that says, "On Yom Kippur the custom is to wear at least something that is white." Not a hard standard to meet, right? You might pull off one item of white just by accident.
Third, especially for the ladies, check the weather report first. On a rainy Yom Kippur, there will be far less white clothing in shul, since women's clothing often become see-through when wet. (Remember: umbrellas aren't allowed on Shabbat or yom tov, so you're stuck with a hooded jacket or raincoat...unless you forget it and get caught in the rain.) I've rarely seen that be a problem with men's clothing, which I assume is a sexist conspiracy designed to keep The Woman down. Of course, you can wear undershirts, especially a non-white undershirt, but speaking from experience, some men will stare at a wet white shirt, regardless of whether or not it's see-through. (Personally, I always wear an undershirt because it feels more tznius as a larger lady. It is not required, though some may tell you different.)
Fourth, watch where you sit. Especially around kids. If you're wearing white pants, don't accidentally sit in nastiness. Kids don't have to fast, so if you sit in some chocolate, it's going to look like there's poo on your pants, and that is going to distract the people sitting behind you. I may or may not be speaking from experience.
G'mar chatima tova and shana tova! May you have an easy and meaningful fast!
Remember to check the eruv before chag! No carrying on Yom Kippur. (We'll talk about why at another time, but trust me.)