Befitting the frontier image, dress is mostly casual day and night. Unless you're on a cruise, pack just one outfit that's appropriate for "dress-up" (though even this one set of nice togs probably won't be necessary).
Not all of Alaska has the fierce winters usually associated with the state. Winter in Southeast and Southcentral coastal regions is relatively mild—Chicago and Minneapolis experience harsher weather than Juneau. But it's a different story in the Interior, where temperatures in the subzero range and biting winds keep most visitors indoors.
The best way to keep warm is to wear layers of clothing, starting with thermal underwear and socks. The outermost layer should be lightweight, windproof, rainproof, and hooded. Down jackets (and sleeping bags) and cotton clothing have the disadvantage of becoming soggy when wet; the newer synthetics (particularly wind-blocking fabrics) are the materials of choice. Footgear needs to be sturdy, and if you're going into the backcountry, be sure it's waterproof. Rubber boots are often a necessity in coastal areas, where rain is a year-round reality. When wearing snow boots, be certain they are not too tight—restricting your circulation will only make you colder.
Summer travelers should pack plenty of layers, too. Although Alaskan summers are mild, temperatures can vary greatly through the course of a day. A lined waterproof coat, light gloves, and stocking cap make those glacier cruises and halibut charters much more enjoyable.
The summer months are infamous for dense clouds of mosquitoes and other biting insects. These pests are generally the worst in Interior Alaska but can be an annoyance throughout the state. Bring mosquito repellent with DEET. Also occasionally used (but less effective) is the Avon product Skin So Soft. Mosquito coils may be of some help if you are camping or staying in remote cabins. Head nets are sold in local sporting-goods stores and are a wise purchase if you plan to spend extended time outdoors, particularly in the Interior or on Kodiak Island.
Wherever you go in Alaska (and especially in Southeast), be prepared for rain. To keep dry, pack a collapsible umbrella or bring a rain slicker, as sudden storms are common.
Always bring good UVA/UVB sunscreen with you on outings, even if the temperature is cool. Sunglasses are also essential, especially for visits to glaciers. A pair of binoculars will help you track any wildlife you encounter.