An aicme is a family of five ogham letters that share one way of meeting the writing line and differ only in how many strokes they carry.
The geometry
Aicme Beithe is cut as strokes lying to one side of the line and not crossing it. Its five members carry one, two, three, four and five such strokes: Beith, Luis, Fearn, Saille and Nion, with the values B, L, F and S and N.
On a standing stone the line is the arris, the edge where two faces meet, and the strokes of this family run out onto one of those faces. In the horizontal layout used by the manuscripts the same family is written below the line.
Why the families are named this way
Each aicme takes its name from its own first letter, in the genitive: the family of Beithe, the family of Uath, the family of Muin, the family of Ailm. The naming is descriptive rather than doctrinal, and it is the manuscripts, not the stones, that record it.
The mirror problem
Aicme Beithe and Aicme hUatha use identical strokes on opposite sides of the same line. Nothing in the marks themselves says which side is which; that depends on the reader knowing the direction of the inscription. Read a stone the wrong way round and every letter of these two families becomes its counterpart in the other, producing a different text rather than an obviously wrong one. The convention of reading upward along the left arris exists to remove that ambiguity.
Twenty letters follow from four families of five, and the whole alphabet is that arithmetic.