Ask veit ek standa,
hár baðmr, ausinn
þaðan koma döggvar,
þærs í dala falla,
stendr æ yfir grænn
(“There stands an ash called Yggdrasil,
A mighty tree showered in white hail.
From there come the dews that fall in the valleys.
It stands evergreen above Urd’s Well.”)
Old Norse was the language spoken by the Vikings, and the language in which the Eddas, sagas, and most of the other primary sources for our current knowledge of Norse mythology were written.
Old Norse is a member of the Germanic family of languages, which also includes English, German, and several other languages that are widely spoken today. During the first several centuries of the Common Era, a distinctly northern dialect of Proto-Germanic (the common ancestor of the Germanic languages) formed in Scandinavia, which gradually morphed into Proto-Norse, which, by 750 CE or so – that is, by the beginning of the Viking Age – had become the language we would today recognize as Old Norse. Over the centuries, Old Norse continued to fragment into more regionally-specific languages, and by the early modern era, it had been transformed into Icelandic, Norwegian, Swedish, Danish, and Faroese.
Even in the period in the evolution of the Scandinavian languages that we identify as “Old Norse,” there were regionally specific dialects. Thus, one often hears “Old Icelandic,” “Old Norwegian,” “Old Swedish,” and “Old Danish,” “West Old Norse,” and “East Old Norse” used interchangeably with “Old Norse.” These dialects were all extremely close. For example, Old Icelandic hringr, “ring, circle” appeared in Old Norwegian as ringr. The differences between them were probably about as significant or insignificant as the differences between modern British English and North American English. Speakers of Old Norse all referred to their language as dönsk tungu, “Danish tongue.” 
Just for the sake of clarity, though, when you find a textbook on “Old Norse,” it’s almost invariably Old Icelandic, since Iceland is the country that produced the overwhelming majority of the surviving literary works in Old Norse. I’ve never even seen a textbook that teaches the other dialects, but they’re probably out there somewhere.
As the Vikings raided and settled in new lands during the Viking Age, so too did their language. At its broadest extent, Old Norse was spoken in Scandinavia, Iceland, the Faroe Islands, the British Isles, continental Europe, Russia, Byzantium, Greenland, and even North America. Several common English words are loan words from Old Norse, including egg, guest, gift, score, trust, anger, and want.
How to Learn Old Norse
If you want to learn about Norse mythology, there are few things that will aid you in your quest more than learning Old Norse. Dabblers will probably have no need of or interest in learning the language of the Vikings, but the more serious a student of pre-Christian Germanic/Norse mythology you are, the more useful you’ll find having a solid knowledge of Old Norse to be. This is especially the case for people whose spirituality is rooted in that of the heathen Germanic peoples; Old Norse is to that spirituality what Latin is to Catholicism, what Hebrew is to Judaism, and what Arabic is to Islam.
When I began teaching myself Old Norse several years ago, there weren’t any especially good or up-to-date textbooks on the language. I did what most people who wanted to learn Old Norse did at the time: cobble together knowledge from several sources of vastly varying quality on Old Norse and modern Icelandic. (Modern Icelandic is about as close to Old Norse as Shakespearean English is to modern English, and the available modern Icelandic textbooks were of much higher quality than the available Old Norse textbooks.) Specifically, I mostly used a combination of Auður Einarsdóttir’s Learning Icelandic, Geir T. Zoëga’s A Concise Dictionary of Old Icelandic, Michael Barnes’s A New Introduction to Old Norse, and the free but very basic and incomplete lessons at Old Norse for Beginners.
Luckily, however, you don’t have to go through that laborious and often confusing process anymore. Archaeologist and historian Jesse L. Byock, the author of the excellent Viking Age Iceland and translator of numerous sagas, has written a highly useful, accessible, and engaging textbook of Old Norse for the twenty-first century: Viking Language 1: Learn Old Norse, Runes, and Icelandic Sagas. (Its companion volume, Viking Language 2, will be released in the near future.) Viking Language 1 includes lessons in Old Norse grammar and vocabulary, brought to life with excerpts from the sagas and informational sections on Viking society. Graded lessons enable you to check your progress.
After buying and perusing this book, I really wish that this or something similar had been available when I first started learning Old Norse. It would have saved me a lot of trouble, time, and money.
When reading an unfamiliar text in any new language, it’s necessary to have a dictionary so that you can look up words you don’t recognize that are crucial to the meaning of a sentence or passage. Zoëga’s A Concise Dictionary of Old Icelandic is still by far the best out there, so I recommend picking up a copy of it along with Viking Language 1.
Old Norse is probably harder than your average language to learn, but studying it is an extremely rewarding endeavor and well worth it. In addition to taking your understanding of Norse mythology and the early Germanic world to a whole new level, you’ll be able to speak one of the most harshly beautiful languages ever spoken and read and appreciate the nuances of some truly breathtaking poetry in its original language. As an added bonus, I’ve met a surprising number of people who think it’s quite sexy.
 The Poetic Edda. Völuspá, stanza 19. My translation.
 Barnes, Michael. 1999. A New Introduction to Old Norse. p. 1-2.
 Byock, Jesse L. 2013. Viking Language 1: Learn Old Norse, Runes, and Icelandic Sagas. p. 20-23.
 See the entries for these words in the Online Etymology Dictionary.