*Disclaimer. I'm recommending items to take when you are moving abroad. Not forever, but also not for a two week trip. I'm assuming you have at least one, maybe two suitcases, and you are staying long enough to need this.
Things you'll regret not taking:
A headlamp. Not a flashlight, but a headlamp. Preferably one that is easy to carry, uses normal AA batteries, and is really bright. I can't tell you how often the power's gone out and I've thanked God for my headlamp.
Stamps and envelopes. I buy a bunch of Forever stamps before I leave and carry them in my wallet. If you are going to living in a more developed area, these may be irrelevant, but since international mailing services are expensive and difficult to find in some places, my biggest source for mail delivery has often been people who were traveling to the US from wherever I was living. They are almost always willing to carry mail, but it's much easier to be able to hand them a stamped, sealed, addressed envelope and just tell them to drop it in the mail.
A sewing kit. Assuming you know how to sew. And if you don't, I'd recommend learning. At least enough to mend your clothing. Your kit can be small, but it comes in handy. It's also really fun breaking the stereotype that Americans don't know how to sew/cook/etc.
Medicine. I know; that makes you laugh. There is medicine everywhere, and a lot of it is cheaper to buy than its equivalent here. But you know that one brand of cough drops you love? The mole skin you use every time you get a blister? You can't always find those overseas, and trust me, when you get sick, you'll want them. Don't bring a lot, but a small med kit of the things that make a sniffly you feel at home... it's worth the weight.
Gifts. These are hard, because you haven't yet met the people you are buying for. But I find that personal is helpful. Things that say the name of your home city- mugs, keychains, even postcards can be fun. Things specific to your country or hometown- I've known a lot of Canadians who take maple syrup with them to share with new host families. In places where there are a lot of expats, they may appreciate gifts that remind of them of the homeland you share. Or gifts that are practical- in Afghanistan last Christmas, the best gifts I saw were from a family who had brought boxes of good matches for all their close friends, because the flimsy matches there drove us crazy every night as we lit the wood stoves.
Duct tape. Enough said.
Rechargeable batteries and chargers. Especially those cool chargers that can do both AA and AAA. Also, remember that your plugging-in situation might be complicated, and a really heavy charger attached first to a converter and then hanging from the wall will break, so pay attention to the direction of the plug when you are purchasing it.
Household items. Tide-to-Go sticks, command hooks, double-sided tape, those little key ring things that open and let you put flashcards on them (please tell me you know what I'm talking about?) Fingernail clippers. Super glue. Dental floss.
Extra passport photos. I buy ten or so at once (At a cheaper place, or photographing and printing my own) and keep them in my wallet.
Ziploc bags. Not always for sale, but so useful. I just buy an extra box of gallon ones before I leave.
Things that could come in handy
Backpacker's towel. I'm editing (Jan 2014) to add this. I bought a medium backpacker's towel for my current trip, and I love it. It's really lightweight, but it is moisture wicking material, so I can dry my whole body and start my hair with a piece of fabric 1' by 2'. I'm definitely bringing this in the future to save buying or bringing a heavy towel everywhere I go.
Exercise equipment. A resistance band and a few workout videos (or whatever equivalent you use) aren't too much extra weight in your suitcase, and if you are traveling to places that may not have much emphasis on sports, there likely won't be a lot of access to sporting equipment.
Black socks. Sounds silly, I know, but I've spent a lot of fall/spring days wearing black sandals and wishing I hadn't packed primarily white socks. Plus, black socks stay (read: look) clean longer.
Water filter. I bought a backpacking water filter years ago that I take with me everywhere I travel, and I love it. I've gotten sick more than once from drinking water in places where sewage systems aren't very high quality, so now I'm very careful about what I drink. You probably don't need this, as bottled water is usually available, but it has come in very handy.
You-specific stuff. Clothing can be purchased anywhere. You can almost always find a pillow or some aspirin or most necessities. The little things that make you happy, however, can't always be purchased abroad. Whether it's art supplies (I always take supplies to make homemade birthday cards) or a brand of toothpaste or a particular food you aren't likely to find (peanut butter is a common contender in this category, but I'm the kind who carries hot apple cider packets with me), I'm betting you'll be thankful you brought it.
Dictionary. You can get them on your device now, or you can buy one there, and those are great ideas. But considering the importance (in terms of safety, of respect for your host culture, of your own feeling of independence) of being able to look up a word on the go, it's useful to bring a small one with you.
Miniature items. I don't like taking shampoo and conditioner with me, but I do like the pocket-sized containers. A lot of places abroad don't offer miniature containers and other items, but they are very useful. If you'll want them, plan ahead!
Work-specific stuff. Do your research. A lot of my traveling has involved education in some way, so I like to know ahead of time what sorts of supplies I can expect to find. Few places I've been have had access to cheap stickers, for example, so I brought a sticker book to Italy, and my students loved it. Electronic timers for debating weren't available in Afghanistan, and many countries don't sell index cards like we do. All of these are things we are used to having at our fingertips, so it's helpful to think ahead and bring a few.
Things from home. The people you meet will want to see pictures of your family, and you may not have Internet to show them Facebook. Bring a mini photo album, a flag, postcards from your area, etc. They'll love hearing about the place you are from!
Boys, look away... ladies, bring tampons. Most places around the world don't have them readily available. I didn't find ones I liked even when I was in Japan, and you know as well as I do that this isn't something you want to adjust to changes with. Just bring a supply and give away what you don't use; any foreigner friends you make there will love you.
Things everyone says you'll want but you really won't
Toiletries- unless you have a specific brand you care about, can't bear the idea of adjusting, and are willing to bring enough to last your entire time. If not, just buy as soon as you arrive and adjust that much sooner.
*My exception to this is deodorant, because I've never found a scent I like when purchasing in a foreign country. In a lot of places I've been, women wear deodorant that smells like men's deodorant or aftershave here, and I've never been able to get used to it. If you can, more power to you!
Clothes. Let's be honest. How much do you know about fashion where you're going? Not much, I'd bet. Take the basics, but leave the multiple wardrobes at home. You can often survive on few clothing items, supplementing your closet with items that are better suited to the climate, the culture, and the fashion.
*One exception here is boots. If you are going to a (developing) place that will want rain or snow boots, you likely won't be able to find good ones for a decent price.
Converters. Again, this will be easier once you arrive. Your fancy-looking one you bought at that travel store or at Radio Shack will get lost, stolen, stolen but everyone will say lost, or left behind at some point. Just buy the cheap one when you get there.
Passport holder. So many people tell me to carry these, but I hate them. Modern women's clothing is not designed to surreptitiously have a passport holder hiding underneath, and when they hand outside, they are an invitation to get robbed. I wouldn't recommend them. (That's my opinion, but there are plenty who would contradict this, so use your own discretion.)
Most of the items I've listed probably aren't top-tier importance, but their presence may make you less homesick and more comfortable, and I think it's worth it. Everyone's suitcase will look different, and I hope you've done enough research to know the important items to take. (Remember- passport, visa, cash, cell phone. Chargers. Prescription drugs in original bottles/contacts/inhaler/anything else you may need. All that other stuff that proves you are a responsible adult.)
Lastly, a word of warning. You want to be comfortable, but you also will need to lug all this around airports, through customs, etc. You'll likely be moving it or unpacking it in front of people in your host country, who will see your mountain of oh-so-necessary stuff through different cultural eyes. So be cautious. Take what you need to live a wonderful life while you are there, but don't be afraid to let go of some of those physical bonds and embrace something new.
But don't forget the duct tape.