an invariant
Some features of a shape survive being moved around; others do not. Slide and spin a triangle on the table and its side lengths and angles stay the same, but its position and orientation change. The properties that refuse to change under the allowed moves are the invariants of that geometry — they are the things the geometry can actually talk about.
Precisely, given a transformation group acting on a space, an invariant is a quantity or property of figures that takes the same value before and after applying any transformation in the group. There are two flavours worth keeping straight. A numerical invariant is a number that the transformation never alters: distance between two points is invariant under rigid motions, since rigid motions preserve length. A property invariant is a yes-or-no feature preserved by every group element: 'these three points are collinear' survives every projective transformation, so collinearity is a projective invariant. The recipe for spotting an invariant is exactly the Erlangen recipe: a feature is invariant for a geometry precisely when it is constant along the orbits of that geometry's group — figures in the same orbit must share it. A closely related but distinct idea is a covariant: a quantity that does not stay fixed but transforms in a controlled, predictable way (a triangle's centroid is not fixed by an affine map, but it moves to the centroid of the image triangle).
Invariants are the whole point of the Erlangen program: choosing a geometry means choosing what you refuse to let change, and the theorems of that geometry are relations among its invariants. As you enlarge the group, invariants are lost — length falls away first, then shape and angle, then parallelism and ratio, until only incidence and cross-ratio remain under the projective group. A common slip is to assume a property invariant under a small group stays invariant under a larger one; usually it does not. Length means nothing to an affine map; do not carry a Euclidean invariant into a geometry whose group is too big to respect it.
The cross-ratio of four collinear points is a single number that every projective transformation leaves unchanged, so it is a projective invariant — the richest one the projective group permits. Distance, by contrast, is invariant for rigid motions but not for similarities (a scaling halves every distance), showing that an invariant is always invariant relative to a stated group.
A property is an invariant only relative to a named group; enlarge the group and invariants vanish.
An invariant of a small group need not be an invariant of a larger one — length survives rigid motions but not scalings, so 'invariant' is meaningless until you say invariant under which group.