An axis is an orthonormal basis-vector, a primary element in a tuning lattice diagram created in some type of pitch-space, typically a prime-space.
Each axis of the lattice typically represents one particular prime number, each subsequent axis representing the next prime and usually excluding 2 ... thus: 3, 5, 7, 11, 13, 17, etc. However, any prime-factors may be present, with some or many skipped over.
Each axis of a lattice typically is centered on the origin, where all prime-factors are to the zeroth power. Negative and positive exponents extend outward in opposite directions from the origin, as equally-spaced steps along each axis.
Each axis must have its own unique angle in order to distinguish properly among the different prime-factors. Many various lattice formulae exist for this purpose, and give each author's lattices their own distinctive appearance. In addition, other parameters may be encoded (and even redundantly encoded) by various other aspects of the axes: length, thickness, color, etc. (For example, Paul Erlich's lattices in his paper The Forms of Tonality use length and color; Joe Monzo's lattices use length and thickness.)
In the cases where the pitch-space is different from prime-space, each axis will represent some other set of musical intervals.