How can emission spectra used to identify presence of specific elements in a substance?

In atomic spectroscopy, each element has a unique spectrum. The atomic spectrum obtained from a sample is a combination of the spectra of each elemental component. We take the strongest line from the sample spectrum and determine which elements could have caused it (we call these "candidates"). We then look at the full spectrum from each candidate and see whether or not every major line is present in the sample spectrum. If so, we say that element is present.

Then we subtract the spectrum (or spectra) of the element(s) we have determined to be present from the sample spectrum and repeat the same process with the next strongest line in the (leftover) sample spectrum. We continue repeating this process until all lines in the sample spectrum are accounted for.