Sure there was Paris and great heathen army (which lost majority of pitched battles) but aside from that I fail to see what's so fierce about killing a monk or unarmed peasant.
I will attempt to answer this with a serious reply.
Vikings (or more accurately, the Norse) did not just attack undefended monasteries and villages. True, that is what they are most famous for, because it was primarily their more literate enemies who popularized this notion, which has now been ingrained in modern-day textbooks. Vikings earned their fierce reputation through constant military success on the battlefield.
They directly threatened inland capital cities of major political states (they laid siege to Paris multiple times). The Norsemen were responsible for carving out kingdoms and polities far from their original homelands, in Britain (kingdom of York), France (Duchy of Normandy), Eastern Europe (Kievan Rus), to name a few. In fact, the founder of the political state that was the predecessor to Russia, was a Norseman (ie Rurik).
Furthermore, the fact that the Norsemen were able to so successfully scatter and subjugate often similarly fierce Germanic opponents known for their martial abilities (ie, Angles, Saxons) repeatedly only added to their reputation. The fighting prowess of the Norsemen spread far beyond the lands they conquered as well, as they were known as far away as in the Middle East, where Norse mercenaries (known as the Varangian Guard) were employed as extremely effective shock troops by the mighty Byzantine military machine. Furthermore, one faction of the Norsemen (ie, the Normans) would go on to conquer England and Ireland, and carve out empires in the Mediterranean (Kingdom of Sicily), for example.
Edit:
To clarify several points of criticism, I want to add:
While it is true Vikings did prefer to attack undefended settlements, one must not forget that they were first and foremost opportunistic raiders with primary objective of obtaining material wealth. However, many Vikings were so impressed by amount of wealth they plundered that some decided to stay permanently in the lands that they once raided. Vikings were not the first, nor the last group of people to raid European coastal shores (the Gaels of Ireland, the Barbary pirates from North Africa did the same). However, the Vikings achieved a lasting impression because they were able to establish durable political entities through victories on the battlefield and shrewd diplomacy.
In addition, one must consider that, true, the average Viking was essentially a peasant known for exceptional violence, and one could chalk up their fierce reputation to that as a result. This however, does not detract from their reputation to make war, as this propensity for violence and their ability to inspire fear in their enemies were traits unique to Vikings for that time period, and as one might ask, "who could imagine the average malnourished European peasant to do the same?"
sources:
Forte, Angelo; Oram, Richard; Pedersen, Frederik (2005). Viking Empires
Raffaele D'Amato (2011). The Varangian Guard 988-1453
Hadley, Dawn (2006). The Vikings in England: Settlement, Society and Culture
In medieval Ireland we can identify two major attitudes towards the vikings/Scandinavian raiders that vary considerably while maintaining core similarities at the same time: the attitudes of the clergy who suffered from raids and who compiled the very sources that we have about the viking period, and the attitudes of the Irish warrior-aristocracy who fought against them, and sometimes even allied with them.
From the perspective of the clergy, the arrival of Scandinavian raiders targeting monasteries was not really as earth-shattering an event as we might imagine today. In fact, Irish people had been raiding their own monasteries for decades before the vikings even set foot in Ireland and would pillage more monastic communities throughout the period of the 'viking age' than the vikings ever would! Local communities might have sacked monasteries during hard times for food and resources, while warrior-aristocrats probably targeted monastic sites in enemy territories, which would have been the only really concentrated kinds of settlement in the island at the time (most people lived in isolated farmsteads scattered around the countryside). Monasteries weren't as passive as we might imagine either and often waged outright wars against one-another: in 760 the monasteries of Birr and Clonmacnoise fought a pitched battle, while in 807 the communities of Cork and Clonfert clashed on the battlefield, ending in “an innumerable slaughter of the ecclesiastical men and superiors of Cork.”
What made Irish monks fearful of viking raids was probably their rapid, amphibious nature and the fact that the raiders were not Christian. However the contemporary reputation of vikings as some medieval super-warriors is probably an invention of the Romanticist era and modern nationalism; as we'll see they weren't particularly successful when confronted by a competent military force. Instead, it was the idea that the raiders could slip off into their boats and disappear across the horizon that would have been scary - if a neighbouring Irish lord came and sacked your altar, you knew where he lived and could organize a counter-raid to try and reclaim the goods. The anxiety of Irish monks towards the rapid amphibious raiding style of the vikings is neatly summed up in this 9th century poem found in the margins of a manuscript:
Is acher ingaith innocht
fufuasan fairggae findfolt
ni agor reimm mora minn
dond laechraid lainn ua
lothlind
The wind is rough tonight
tossing the white combed ocean.
I need not dread the [fierce warriors of Scandinavia]*
crossing the Irish Sea.
From the perspective of the Irish warrior-aristocracy, the arrival of the Scandinavians meant something completely different. By this period, Irish society would have been considerably more militarized than in the previous centuries, as provincial kings began to wage wars of hegemony over neighbouring territories which accompanied a lower-level kind of endemic warfare in the form of cattle-rustling and retaliatory raids. This is a period where the Irish conception of kingship begins to shift from a mostly ritual and priestly role to a cold rationalization that sovereign authority could be derived from military might. Needless to say, it was actually pretty bad timing for seasonal Scandinavian raiders to set foot in Ireland: they had entered an arena of unending warfare where regional warlords were making far-ranging mobile campaigns into enemy territories.
While raiders could make amphibious assaults on monasteries and communities, oftentimes they still had to haul their loot a considerable distance or even camp out the night before scuttling off, in which time they were the perfect prey for an Irish warlord leading a small, semi-professional army nearby. Annalistic evidence attests to the fact that wandering viking parties were sometimes confronted and slaughtered by more mobile Irish warbands who would then take the raiders' spoils for their own!
However the aristocracy's view of the vikings was not totally negative: as I've said, this is a period where regional warlords are vying for military and political supremacy. Consequently, the introduction of fresh fighting men and Scandinavian weaponry and tactics was seized by some lords as an opportunity to increase their own military potential. Barely a few decades after the first raids, viking bands are recruited as mercenaries by Irish kings. The Fragmentary Annals of Ireland contains one particularly vivid example, where Cerball mac Dunlaige, King of Osraige, played rival raiding parties against one another; as his territory was being plundered by a band of Norwegians, the king made a hasty alliance with a band of Danes who fought alongside his army and slaughtered the Norwegians. With the establishment of permanent Scandinavian coastal settlements, Irish lords had more permanent access to military hardware and manpower, and greatly enriched themselves by participating in the Scandinavian slave-trade.
Afterwards, those Danes remained part of Cerball's army in his wars against other Irish kings and Norwegian raiders, although the text is clear that this arrangement was always shaky - in the context of early medieval Irish warfare, today's allies could easily become tomorrow's enemies. The attitude of the aristocracy towards the vikings is kind of paradoxical; on one hand, they are a welcome asset to bolster a lord's army, but on the other they are viewed as cowardly and prone to fright (which may be a commentary on their status as pagan foreigners in the context of the Irish status system - this would have made them essentially the lowest-status persons in the island and would have been expected to show fear before a high-status Irish king). Annal entries describing confrontations between the two frequently contain a phrase that goes something like this:
When the Norwegians saw Cerball with his army, or retinue, they were seized by terror and great fear...
Whether this reflects the actual emotional reactions of the vikings to approaching Irish armies or the expected roles to be played by a low-status individual in the face of a high-status king is uncertain, but I believe it's a mix of the two. So the attitudes of the early medieval Irish towards the vikings depended on one's place in society. For the clergy, vikings were sort of scary because they could strike and escape at any time, but their reputation as monastery-sackers shouldn't be overplayed considering the long-lasting tradition of Irish monastery sacking. For the warrior-aristocracy, the vikings were paradoxically a welcome addition to their own military power when they could be allied with, but if not were seen as cowardly, fearful pagans who could be preyed on for their spoils.
I'm not saying this is a recurring troll thread or anything, but to quote Cowboy from Full Metal Jacket, "I've seen this before!"
There's numerous good answers in this thread, and I think /u/thatchairman basically does the job. If you follow the above link, I gave some information ripped from a textbook on the exact same question, and also made the point that they did not lose the "majority of pitched battles." I think /u/thatchairman probably provides better sources however (Edit: Apparently not according to /u/Aerandir). I stand by my original point that this thread seems to recur with similar wording for some reason, almost as if it's the same person posting it or something.
Because they didn't only attack undefended monasteries and villages, and the Great Heathen Army didn't lose the majority of pitched battles. "Sure there was Paris" and yet the Viking invasion of Western Francia was not a singular event. The singular event of Paris was in 845, and major Viking expeditions devastated Western and the Middle Kingdom for decades after that. In 882, for example, there was a great raid to Cologne and Trier. And this is only Francia here. The extent to which the Vikings subjugated Britain was even worse. Kievan Rus also shouldn't be forgotten - it even besieged Constantinople.
Sources: The Northern World in the Dark Ages, 400 - 900. -Edward James