Item 5 here has a figure calling Lobachevskian geometry hiperbolic and Riemann geometry elliptic and both figures have a perpendicular line.
What does that perpendicular line and the both pictures mean?
Does that mean that any 2 parallel lines from Euclidean space always become parts of some hyperbola or ellipse in Lobachevsky or Riemann space respectively ?
The perpendicular lines are meant to suggest the essential differences among the geometries when considering lines through a point parallel to a given line. The curves need not be part of actual hyperbolas or ellipses.
In Euclidean geometry there is exactly one one such line - and a line crossing both parallels and perpendicular to one will be perpendicular to the other.
In hyperbolic geometry there will be multiple lines through a given point parallel to a given line. The intuition behind the picture is to imagine other hyperbolas in the top half of the picture, arranged so that they do not intersect the one in the bottom half.
In elliptic geometry there will be no parallels. The picture showing the lines curving toward each other suggests that. Think about two great circles of longitude intersecting the equator at right angles.