I personally chose 'four' to write about on a whim, because 'cuatro' sounded different enough, but was relatively explainable in a paragraph. But this only goes back to languages in the Indo-European tree.
The paper addressed here says that these 'ultraconserved' words share an ancestor before Indo-European. Via page two:
In addition to Indo-European, the language families included Altaic (whose modern members include Turkish, Uzbek and Mongolian); Chukchi-Kamchatkan (languages of far northeastern Siberia); Dravidian (languages of south India); Inuit-Yupik (Arctic languages); Kartvelian (Georgian and three related languages) and Uralic (Finnish, Hungarian and a few others).
So, if you can imagine doing that process we did, but backwards, trying to take all these hundreds of languages, living and extinct, in the Indo-European family [1] and making one giant proto language that they all descended from, with a bunch of rules that determine how they morphed to become the child languages, then you get Proto-Indo-European. Now do that again for the other language families above. Then compare the resultant hypothetical languages, and see if
they share enough between them to say that they are all children of the same language. This is a lot more controversial, see [2] (what this paper seems to subscribe to) and [3].
What the hypothesis seems to be here (this is speculatory; I haven't read the paper) is "If we look at these proto languages, there seem to be words that are similar enough that they probably are the last remnants of some ultra-proto language that all of these language families descended from".
And all words are coined and undergo evolution at some point, but often the popularity of words for the same thing can change, and not all words come from antiquity.
Inventions like 'piano' and 'internet' obviously haven't been around very long in the grand scheme of things. But we have other words like 'cater' and 'embrace' that are attributed to Shakespeare [4].
Some words that haven't been conserved: leafworm for caterpillar, wig for war [5], clepe for 'to call', except in the nearly-dead yclept [6], which also contains the ge/y prefix that makes past participles [7].
If you check out [8], you'll spot a bunch of words you probably know, but with senses they don't have anymore.
So words are always coming and going, and it seems that this paper is trying to say that certain prevalent words today statistically seem to be from a time before PIE. This reeks a bit of glottochronology [9] to me, though, which is fairly controversial.
[1] http://upload.wikimedia.org/wikipedia/commons/4/4f/IndoEurop...
[2] http://en.wikipedia.org/wiki/Eurasiatic_languages
[3] http://en.wikipedia.org/wiki/Proto-World
[4] http://www.pathguy.com/shakeswo.htm
[5] http://en.wikipedia.org/wiki/Changes_to_Old_English_vocabula...
[6] http://en.wiktionary.org/wiki/yclept
[7] http://en.wiktionary.org/wiki/y-
[8] http://en.wiktionary.org/wiki/Category:English_archaic_terms
[9] http://en.wikipedia.org/wiki/Glottochronology