The Asian tiger mosquito, an invasive species, was first found in the continental U.S. in August, 1985, in Harris County, Texas, in significant numbers. Two researchers published an article about the discovery in the Journal of the American Mosquito Control Association.  The next year, more tiger mosquitoes turned up around a pile of used tires near Jacksonville, Florida.  Since then, they have been spreading rapidly, from both new arrivals and breeding.
Even before they took up residence in Texas, a single female of the species was found in Memphis, Tennessee in 1983, so the Asia mosquitoes had probably been coming to the United States, hidden in cargo, for a while. In Hawaii, tiger mosquitoes had already moved in as early as 1902. 
The first few mosquitoes probably arrived as dormant eggs in shipments of used tires, but since then, the invaders have spread to at least 26 states, according to the Centers for Disease Control, and have been found as far north as Chicago and Minnesota. They prefer warm weather, so they have been identified in every county in Georgia, Florida, South Carolina and Tennessee, as well as many counties in surrounding states. They breed in small puddles of standing water near human habitation, in diverse places, varying from cemetery vases to junk piles.
Like other invasive species, the Asian tiger, Aedes albopictus, has out-competed mosquitoes who already lived here, such as Aedes aegypti, the yellow fever mosquito, but that's not necessarily good. The tiger mosquito can spread diseases too, such as dengue fever or Eastern equine encephalitis. In fact, its habit of staying close to where people live and biting multiple hosts in the daytime, make it an even more efficient vector for some diseases. We're lucky that it came to the United States as uninfected eggs, rather than as infected adult mosquitoes, and that not enough people or animals infected with the diseases it spreads are available here, so no epidemics have started. Yet...
After the first arrivals, the next mosquitoes were found in a pattern that matched U.S. highways, hinting that the eggs or larvae, or perhaps even the adults, were being transported throughout the U.S. unknowingly, probably also in used tires. In 1988, a Public Health Service Act started requiring any shipments of used tires to be fumigated, if they came from a country that had tiger mosquitoes. Unfortunately, the law might have been too little, too late, since once the mosquitoes hatched in the U.S., they had already found plenty of places to breed and spread here. 
This series of maps shows how the invaders spread during the first ten years after their arrival. The newly discovered populations are shown darkest in each map, with the older populations in gray.
By now, tiger mosquitoes live in most of the southeastern United States, as shown by the orange areas in the map below. Other places, such as California, have also reported invasions which officials are trying to eradicate before they become permanent.
So how can you tell if you've actually seen an Asian tiger mosquito? One thing to look for is the black and white stripes, which gave it the name "tiger." The stripes occur not only on the legs, like some other mosquitoes. They also show boldly along the body.
It's also more likely to bite you in the daytime. Most mosquitoes rest in the day and come out near dusk or dawn, or at night, but tiger mosquitoes look for prey and bite in the daytime, too, especially in the afternoon. Like other Aedes mosquitoes, they die off in cold weather, leaving eggs near water which hatch again in the spring, producing larvae which turn into new adult mosquitoes in less than two weeks, sometimes as short as a week in hot weather. In subtropical climates, such as southern Texas and Florida, the adults are active all year.
Asian tiger mosquitoes can transmit diseases to humans, but have not yet become a significant vector in the United States. They can carry several kinds of encephalitis viruses, as well as dengue and Cache Valley virus, though other moquitoes do too, and no significant outbreaks have occurred in the United States because of the new invasion. Scientists are watching the situation, monitoring the spread of the mosquitoes as well as testing them to see if they're carrying any disease.
As one might expect from the name, Asian tiger mosquitoes originally came from southeast Asia, but they have now spread to diverse parts of the world, showing up in Africa, Europe, North and South America and Australia. Some areas have prevented them from naturalizing due to careful detection and eradication programs, but they have become established elsewhere, as show by this map. Original habitat is in blue; places it has invaded and become naturalized are in green.
A good site for the latest information is the listing at the USDA National Invasive Species Information Center.
 "The discovery and distribution of Aedes albopictus in Harris County, Texas"
 "Asian Tiger Mosquito," University of Florida
 "Aedes albopictus in the United States: Ten-Year Presence and Public Health Implications"
 Moore & Mitchell, "Aedes albopictus in the United States: ten-year presence and public health implications," 1997
Photo of mosquito on finger from freedigitalphotos.net . Public domain photo of eggs from Wikimedia Commons. Photo of mosquito biting from the CDC.