In Yggdrasil's terms, a Virus is an program which propagates through various different branches, and usually results in incredibly deleterious effects for everyone involved, including causing instability on a given branch. The types of viruses are numerous, and putting a stop to them is a full-time effort for many an Admin, especially Hephaestus. While their effects have been shown to primarily affect Loops internally, the occasional virus has been known to break free of the lower level of Yggdrasil's code and become an even bigger concern for the Admins themselves - in that they will sometimes find themselves attacked or harassed by the entities created.
While the sources of viruses are nearly as varied as the viruses themselves, one source pops up on a regular occasion: The Great Old Ones, described by Sleipnir as the equivalent of script kiddies and trolls. As shoggoths and the like can quickly prove detrimental to the sanity of those who encounter them, this is a direct attack on the stability of Yggdrasil itself, particularly if a Loop Anchor finds their mind eroded by the presence of those viruses. In return, the admins created a unique antiviral program to deal with that side of the problem: Old Man Henderson. While almost as virulent as the Lovecraftian viruses, it's nonetheless much less harmful to the mental state of Loopers... mostly.
One extremely harmful virus, or possibly a collection of them, was responsible for corrupting the data of the Megaverse, which - upon attempts to activate the branch - was then the major cause of the event known as The Crash. Albert Wily's code, despite the best efforts of Hepaestus to clean things up, remained deeply affected by the malicious program, rendering the villain in part something like a Dreamer or a Loop-Aware Entity, and it's suspected that the effects of that virus are still preventing otherwise-viable Loopers from being activated.
Sometimes, Admins can be spurred onto actions which would otherwise be considered rash due to the presence of viruses. When Zeus' collection of ... NSFW material was stored on Sleipnir's server and the innumerable pieces of malware there infected Equestria, the resultant Loop led Sleipnir to immediately create an avatar, hide himself and Twilight Sparkle away, and subsequently crash the loop. The damage done to Yggdrasil by that action would absolutely have been less harmful than the mental damage caused by what the Admin described as a 'mad mad mad mad mad world'.
Less harmful viruses are sometimes considered by certain Admins, particularly those who take on the role of tricksters, to be more amusing than harmful -- though they still do their very best to clean them up. For example, the Squirrel Virus affected multiple branches, but simply replaced certain objects with squirrels: Apollo, for instance, found himself with a cup of squirrel. (The Great Old Ones were rather bemused by this, as none of them had come up with it: rather, it was a prank by a pair of lower-level Admins, and easily cleaned up.) Similarly, even Skuld had to admit that the Bee Virus, which bedeviled numerous admins and had previously been confined to the Summerset Isles, was at least amusing.
Within the Loops, several Loopers have developed their own viruses, either as a hobby, for attacking a recurring enemy, or as an active counter to artificial intelligences. While frowned upon by Admins, they by and large have no means by which to harm Yggdrasil. Most of these are not likely to harm a Looping artificial intelligence, as countermeasures are put into place: SkyNet, for example, is unlikely to be afflicted. There are exceptions, however. Pinkie Pie, upon being tasked with creating a program to stop the malicious entity 'CelestAI', created a program for which there is no known defense. If used on an artificial intelligence, it is entirely possible that this could result in permanent damage to their code... a situation which would be made much, much worse if used on a Looping AI, let alone an Anchor.
Another example of an in-Loop virus which could be incredibly harmful to a Looper, even a non-AI one, is a Jain Node, from the Culture loops, as it is not unprecedented that a Looper could find themselves mentally assaulted in such a way that the viral nanotech would be put into their Subspace Pocket, reinfecting the Looper every time, even after a reset.