On the sphere, points are defined in the usual sense.
The equivalents of lines are not defined in the usual sense of "straight line" but in the sense of "the shortest paths between points" which is called a geodesic.
On the sphere the geodesics are the great circles, so the other geometric concepts are defined like in plane geometry but with lines replaced by great circles. Thus, in spherical geometry angles are defined between great circles.
The spherical geometry is the simplest model of the elliptic or Riemannian geometry, in which a line has no parallels through a given point, and it is opposite to Lobachevskian or hyperbolic geometry, in which a line has at least two parallels through a given point.
An important related geometry related to that modeled by the sphere is called the projective plane; it is obtained by identifying antipodes (pairs of opposite points) on the sphere. Locally, the projective plane has all the properties of spherical geometry, but it has different global properties. In particular, it is non-orientable.