How do they identify celestial objects?

Surveys of the sky have been recorded over the course of several centuries. Objects are plotted by Right Ascension and Declination in a format analogous to latitude and longitude on the ground, except that the "longitude" is in 24 hours instead of 360 degrees since the earth rotates in 24 hours.

If the RA and DEC is known (and if the view or image has the direction of North known) then it is possible to identify stars previously plotted.

I recommend downloading the free planetarium software Stellarium. You will learn a thing or two.

http://www.stellarium.org/

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Added: Your additional details make for a completely different question. Some stars won't be visible hundreds of light years away. The Sun would not be visible without a telescope.

From Earth we can plot in 3d the nearest stars pretty well using parallax, up to a hundred light years or so. The closest stars have a measurable proper motion as well. That is, it's motion around the galaxy relative to our own sun. Spectroscopy can also measure radial velocity relative to us.

Beyond that, unless you have standard candles like Cepheid variable stars or type 1a supernovae a star's distance can only be approximated by its physical properties measured by spectroscope or by the presence of a companion star.

There are some other planetarium programs like Starry Night, I think, that will portray our solar system viewed from another star system. Not Stellarium though.

From a photograph, no. they all look the same.

But in a telescope you compare its position to a star chart or read its co-ordinates from the telescope mount.

The word astronomy (astra = star, nomy = numbers) means writing down the star positions in a catalogue

Why would anybody show you a star photo? it's just a white dot!

They have different machines and things which can measure their distance from earth, their radius, weight, amount of radiation being let off etc. So obviously if ones radius is 2000 miles, and the others 2200 miles then they can tell them apart.

Look up star maps on your computer.

i think they can differentiate each star by how much radiation each gives off, brightness of each, size of each, and observing the patterns of it's movement.

I look up.