Immortality
In Fifth Edition Ars Magica there are several ways in canon a mortal (in particular, a magus) can obtain immortality.
First, it must be clarified that immortality means "unaging" rather than "undying". Immortal magi are generally vulnerable to destruction, although this may be more difficult in some cases.
Second, "true" immortality also implies immunity to Warping, obtained by acquiring a Might score. A character could, in theory, remain active with any amount of warping, but in practice the line appears to assume that a Warping Score of 10 will result in Final Twilight through accident or a botch, and sooner rather than later. Lesser immortality allows magi immunity to aging at the cost of warping, meaning that they will exit the game when they become sufficiently warped.
Principles of Immortality
Several principles seem to frame the design of immortality in ArM5. Although not always adhered to, they seem to expalain much of the mechanics.
The eternal does not change Perhaps the principle guiding principle, it seems that those who achieve immortality also find it difficult to change. Change (character advancement) requires external resources, generally raw vis, and is slower.
Freedoms have costs Many ways of obtaining true immortality limit you in some ways. The more free you are, the harder it is to obtain that immortality. 'Hard' here primarily means requiring more magi to cooperate more in your service.
Safety has costs Immortality does not mean invulnerability. The more safe you are, the more it costs - principally in terms of freedoms and effort.
Lesser Immortality
The most common form of lesser immortality is the Hermetic Longevity Ritual. At the canonic power level, Hermetic specialists will be more than capable of constructing potent longevity rituals that will not (or rarely) fail regardless of age. Magi are generally presumed to die of other causes or reach Final Twilight, not to die of old age.
Some magi may choose other paths, due to political reasons (e.g. wanting to make their own ritual) or economic ones (e.g. being unable to afford the high raw vis costs involved). One of these is the Immortality of the Forest, a fruit of a Forest Path presented in Guardians of the Forest. The virtue functions much like a potent longevity ritual, providing immunity to aging at the cost of warping - but at the cost of limiting the magus to the particular forest.
Other magi seek out the Guardian of Nature Mystery. Like the Immortality of the Forest, becoming a guardian limits you to one locale. However, it has the added cost of forbidding laboratory work and severly limiting your character advancement. As a means to power, then, it is highly ineffective.
Finally, Goetic Binding allows one to obtain a sort of lesser immortality. This is generally inferior to the Hermetic longevity ritual, as it requires more investment in its maintainance. However, this path is open to non-magi.
False Immortality
Perhaps the most powerful true immortality in canon is that achieved at the final station of the Criamon Path of the Body. Obtaining full immunity to both aging and warping, the magus is nonetheless free to roam yet also to learn. However, roleplaying considerations limit its application so much that for all intents and purposes this is a way to retire that character, not to persist as an immortal character.
Another sort of true immortality seems to be becoming a Great Beast of House Bjornaer. The Great Beasts certainly seem immune to warping and aging, and to retain enough intelligence to nudge the House and act as Mystagogues. However, it isn't clear they pursue character advancement if at all, why they don't take a more active role, and where they roam. It seems the role of Great Beast is intended to be an NPC role, and not a true path to immortality.
True Immortality
Perhaps the most potent path to true immortality is Becoming, transforming the Merinitia magus into a faerie creature. By not completing the transformation, the magus may become immune to aging without incurring warping, and at relatively small costs. Even if this loophole is not allowed, a magus that fully undergoes the transformation is still totally free to roam Mythic Europe and can advance relatively easily - his magic, however, is seriously impaired, and advancement is still slowed.
Another powerful kind of immortality is the alchemical Elixir. Magi who achieve immortality in this way are free to roam the world, yet their magic is unhindered. Their advancement, however, is somehwat more difficult and (typically) bound up to their Talisman. It is very difficult to achieve.
The Mystery of the Living Ghost provides an alternative - greater safety borne out of the ghost's incorporeality, but at the cost of being somewhat more vulnerable outsdie the ghost's Haunt. It is also much easier to attain, not requiring the cadre of assistant-wizards that the Elixir or Ascendancy require.
Perhaps the most difficult to attain, and certainly the most safe, is Ascendancy to the Hall of Heroes. In some respects it is the weakest choice, as your ability to affect the world and advance is dependent on the actions of other magi. Without a cult to summon its aspects, an ascended magus is unable to affect the world or the saga.