It's possible to have two lines in hyperbolic space which intersect: however, lines can also be "ultra-parallel" and diverge (travel away from each other) which is not possible in a Euclidean space. This is because hyperbolic geometry naturally curves "outward", while spherical geometry curves inward (so *all* lines intersect) and Euclidean geometry doesn't curve at all (so in 2 dimensions, parallel lines stay equal distance apart, and all other lines intersect).

If you put a pen down on hyperbolic geometry and draw 2 lines with a third line intersecting both at 90-degree angle, and then draw straight in either direction, they will diverge and look similar to the ASCII art below (though properly curved, which can't be represented in ASCII). The lines can diverge even if the angle in the middle line is slightly less than 90 degrees.

``` 
\    /
 \  /
 |  |
 |--|
 |  |
 /  \
/    \
```

If you repeat the same procedure on Euclidian geometry, with 90 degree angles the lines will never intersect, but also never grow further apart, like below. And with acute angles, if you draw far enough they will always intersect in one direction.

``` 
 |  |
 |  |
 |  |
 |--|
 |  |
 |  |
 |  |
```

The defining property of hyperbolic geometry, the equivalent version of Euclidean geometry's **parallel postulate** is

-  For any line $R$ and any point $P$ which does not lie on $R$, in the plane containing line $R$ and point $P$ there are at least two distinct lines through $P$ that do not intersect $R$.

This is the the only different property between hyperbolic and Euclidean geometry, and causes all of the other differences like warped perspectives and space seemingly expanding as you travel outwards.

For more info, see https://en.wikipedia.org/wiki/Hyperbolic_geometry
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