In an original formulation of Einstein’s general relativity a gravitation field is represented by a metric on (curved) space-time which satisfies Einstein’s equations. An observer was automatically determined by a choice of a coordinate system. A notion of the observer has evolved since then; nowadays there are a few optional ways of looking at that problem: word-line of a point particle (V. Bolos), congruence of such lines, i.e. field of instantaneous observers represented by a unit vector field – an arrow of time (J. Ehlers), tetrad field i.e. an orthonormal frame of vector fields, etc. In the presented work we were dealing with the notion of the observer as an one-dimensional time-like distribution on space-time represented by the normalized vector field (arrow of time). Therefore, our observer is independent of the gravitational field and can be defined independently of the coordinate system. It turns out that a pair: the metric and the vector field determines a geometrical object on a space-time manifold which is called almost-product (Lorentzian) structure. Its properties allow to categorize pseudo-Riemannian manifolds with such structures. It turn out that we have two main classes of the observers: inertial (geodesic) and accelerated ones. Further calculations showed that there are 8 subclasses for each of them which are related to the geometry of the corresponding three-dimensional orthogonal distributions; e.g. integrable (foliation), umbilical, minimal. We have associated above mentioned geometric characteristics with the ones introduced by J. Ehlers (acceleration, rotation, shear, expansion) which describe the dynamics of the observer.