To look at things in the sky\
Astronomy
No, digital telescopes don't exist. The telescope which we know are optical telescopes and what NASA uses are radiation based telescopes, or something related to radioactivity. As far I know, I've never heard about digital telescopes.
A reflecting telescope.
all telescopes use a mirror.
It is an observatory.
Either with our eyes, or with special instruments such as binoculars or telescopes.
An observatory is the name of a place that uses telescopes and other scientific equipment to study space and astronomy. The Mount Polamar Observitory in the west of the US is an iconic example. Today, Earth based observitories are supplimented by space telescopes.
An astronomer uses telescopes to study space. They observe celestial objects like stars, planets, galaxies, and other phenomena to understand the universe's properties and behavior.
Magnifying mirrors, make-up mirrors, telescopes
Observatory or astronomical observatory.
NASA?
An observatory.