Col ceathrair = Prohibition of four = cousin
Col means prohibition, referring to the prohibition of marrying people closely related to you.
The number (in the case above, four) refers to the amount of people between you and the individual, where you pass up and down along generation and sideways between siblings:
You => Parent => Parent's sibling => Your Cousin
Second cousins are "Col seisir" = "Prohibition of six":
You => Parent => Grandparent => Grandparent's sibling => Their child => Your second cousin.
It seemed to me when I was hovering over the translation that it was implying I have four new cousins, the hover translation doesn't seem to explain the prohibition of four (as in not to marry in the third degree of kindred) it didn't seem to imply a first cousin at all, in fact when it marked me wrong, the translation just said "I have a new cousin" not "a new first cousin."
Sorry, you are right. 'ceathrar' for people, 'ceithre' for things.
As for 'col ceathracha' I'm not sure. The lesson on numbers doesn't specify if you the singular form of a noun following human conjunctive numbers like you do for general conjunctive numbers barring some special cases.
They do give 'cúigear fear' as an example so it appears you do use the singular form, but they also give 'beirt bhuachaillí' so I don't know does 'beirt' mean you use the plural form as well as causing lenition on the following noun.
Wouldn't mind some clarification on this.
If you look up 'triúr' in the dictionary, it gives: '1. (Of persons) Three. ~ fear, ban, múinteoirí, three men, women, teachers.' There are similar examples under 'ceathrar', 'cúigear' etc. So it looks as though the genitive plural is used. But I too would appreciate any clarification of the matter.
According to the 2012 Caighdeán, when using ceathrar as “four (people)”, “cousins” would need to be put into the genitive plural, so it would be either ceathrar col ceathrar (for col ceathrair as “cousin”) ceathrar col ceathracha (for col ceathar as “cousin”). When using ceithre as “four” (which is an option even with people), “cousins” would be in a lenited nominative singular, so it would be either ceithre chol ceathrair or ceithre chol ceathar.
From what I can figure out both are correct it is just a variation in spelling. Apparently col ceathrar is the connacht spelling for col ceathar, however being brought up with Munster Irish I would have always used ceathrar and only finding this mentioned on wikipedia I'm not too sure. http://en.wikipedia.org/wiki/Connacht_Irish#cite_ref-1
col ceathrair is the genitive singular
'Col ceathrair' is the form that makes the most sense to me, with 'ceathrar' in the genitive, following the explanation given above by AnLondDubhBeag and the fact that the genitive is also used for the more distant degrees of kinship (cúigir, seisir, seachtair, ochtair).
However, the plural - col ceathracha - is a bit surprising. I would have expected something more like 'colanna ceathrair', similar to English plurals such as sons-in-law or courts martial, but evidently the phrase is treated as a single unit. Perhaps this was a later formation.