Frame 31 of File:Animated Lorentz Transformation.gif. The Lorentz Transformation is capable of intelligible expression in only one or a limited number of ways, but if this particular animation is copied, I'd just as soon be credited, so tag with CC-by-sa 2.5.

The line of dots that cross the horizontal axis represent events that appear simultaneous in one of the reference frames. The lines parallel to these (green if you look closely) are lines of constant time, (lines of simultaneity) in that original frame.

The line of dots that cross the vertical axis are events which occur in the same place at different times in the same reference frame. The lines parallel to these (a little bluish, if you look closely) are lines of constant position, representing the positions of stationary objects in the original frame. These lines are also known as worldlines.

The animation shows when and where those events would occur from the reference of other frames.

The hyperbolic arcs merely show the arcs of where different observers at the origin would observe the events denoted by the large dots.