Firstly, while we today tend to refer to the population of Scandinavia as a whole as "vikings" today, in old Norse, the word referred only to the people who went raiding or trading overseas - "fara í viking", or "to go on a viking", (a phrase that appears in the Icelandic sagas) meant to go across the sea. The word stems from the word "vik", which means bay or inlet.

Speaking of words and where they come from, the language spoken would have been various dialects of Old Norse - one dialect of which was brought to Iceland with settlers in the 9th century, and while it is not identical, it is similar to today's Icelandic. The Icelandic Sagas can, with some modernised spelling and footnotes, be read by someone who knows modern Icelandic.

A lot of what we know of the overseas raiders come from contemporary written sources from outside of Scandinavia, as Vikings as a rule did not write books. The Viking writings we do have tend to be runestones, comemorating big events - such as people returning from a successful voyage, or the death of someone important. These runestones can be found in various places in Scandinavia, standing in fields or forests (there's even rune-grafitti in the Hagia Sofia!). Common inscriptions tend to be along the lines of "Halvdan, son of Rolf, raised this stone in memory of his father who died in battle", etc., etc.

Seeing as how almost all of the contemporary written sources are foreign to the culture itself, they tend to be biased - especially since they were writing about the raiders, rather than the ones who stayed at home in Scandinavia. The Anglo-Saxon chronicle describes the Viking raid on the monastery at Lindisfarne like so:

"In this year came dreadful forewarnings over the land of Northumbria, terrifying the people most woefully: these were immense sheets of lightning and whirlwinds, and fiery dragons were seen flying through the sky. A great famine soon followed these signs and not long after in the same year, on the sixth day before the ides of January, the harrowing inroads of heathen men destroyed the church of God in Lindisfarne by robbery and slaughter.”

Our knowledge of what they looked like and how they lived is pieced together through contemporary accounts, carvings and statues from the period, as well as burials and archeological finds. Ibn Fadlan, who travelled among the people living in what is now Russia, but who were of Viking descent, describes their appearance in these words:

I have seen the Rus as they came on their merchant journeys and encamped by the Itil. I have never seen more perfect physical specimens, tall as date palms, blond and ruddy; they wear neither tunics nor kaftans, but the men wear a garment which covers one side of the body and leaves a hand free. Each man has an axe, a sword, and a knife, and keeps each by him at all times. Each woman wears on either breast a box of iron, silver, copper, or gold; the value of the box indicates the wealth of the husband. Each box has a ring from which depends a knife. The women wear neck-rings of gold and silver. Their most prized ornaments are green glass beads. They string them as necklaces for their women. Ibn Fadlan, on the Rus merchants at Itil, 922.

Much of contemporary viking art tends to depict men with some manner of facial hair - often beards of various types, while they depict women with long hair, often in braids or knots (these images are variously of statues, ship carvings and decorations in stave churches).

As for societal structure, the Edda-poem Rigsthula (the lay of Rig) gives an account of how the societal structure came about, and describes a society consisting of jarls (the ruling class), karls (free men, farmers and craftsmen) and thralls (serfs or slaves). The poem also depicts the thralls as being dark, swarthy and ugly, the karls as being fairer and more pleasant, and the jarls as being pale and tall and handsome. There is some debate about when the poem was written, but the consensus seems to be that it is a fairly accurate representation of how viking age society functioned. The Rigsthula also wraps up with a description of how the youngest son of the first jarl becomes a king, which implies that in that particular case, the rule of primogeniture did not apply - though how true that was for the rest of viking society, I do not know.

Sources: Danish History Museums article on Viking written sources

A summary of Ibn Fadlan's writings, courtesy of Judith Gabriel

Wikisource has the text of both of the Eddas