Light distorsion/ilusion when looking at bright objects What's the name of the phenomenon when you look at a bright object (e.g. the moon) and see some perpendicular "beams" going top-down and right-left (bare eye observation, also slightly visible on iPhone pic)? 

https://ibb.co/j2b4AF
see the link for a picture
It's a bunch of different phenomena, somewhat related, but still distinct.

Naked eye

Since you mention naked eye observations - seeing a small amount of short spikes around bright objects is normal. The causes can be multiple: diffraction of light in/around your eyelashes, dirt/secretions on the cornea, just regular aberrations in the eye's optical system, or in extreme cases it might actually signal vision problems.

Your iPhone picture

It appears that you're taking the picture through some fabric or some other kind of rectangular network-shaped material. In this case, the spikes are due simply to reflection of light on the individual threads. If you removed all fabric and kept only one thread without moving it at all, the light would reflect on it exactly in the same place.

Telescopes

Many reflector telescopes show a pattern of cross-shaped spikes around stars. Those are due to diffraction. Those telescopes have a secondary mirror supported by an X-shaped element called the "spider". Light is diffracting on the spider and creates a similar pattern of spikes around stars.