Translation:I lose money when I run with my son.
Nuair is really a contraction of an uair ("the hour" or "the time"). It's technically a noun and not a conjunction. To use it as a conjunction for subordinate clauses like in this type of sentence you need the relative pronoun/particle a to introduce the verb and noun of the subordinate clause (in this case, rithim). You can think of it as roughly saying in English, I lose money (at) the time that I run with my son.
Nuair will be called a conjunction by itself in contemporary grammars and dictionaries, but historically and syntactically it's not--it's a noun that when combined with the relative pronoun a forms a conjunctive phrase. So yes, you do need the a.
I'm still confused as to when I would use "leis" rather than "le" I was using "leis" as "with him" before, but then DL got me for not using it as "with it" in a sentence where 'book' was the object. I understand that it is often used in a way that would be unstated in English, but that's about all I've got at this point.
Leis can be a prepositional pronoun, which represents le + é (i.e. “with him” or “with it”). Leis is also used as a simple preposition instead of le when it is followed by a definite article — e.g. leis an leabhar or leis na leabhair rather than *le an leabhar or *le na leabhair.
Caol le caol agus leathan le leathan. Ok, so in the word airgead we have i before rg and e after it, they are both slender vowels, which respects the rule. But, what makes them necessary here then, since we also have a in both sides? I mean, couldn't it be written like argad? I'm not sure if I completely understood this rule =\
Technically air- in airgead is a diphthong--a vowel sound made by combining two vowelss. The first syllable should sound more like air than ar although the dialectal variations in Irish really blur this. But historically it's a diphthong, if you pronounced it clearly as the diphthong ai and not something between ai and a you would be correct and still be understood despite what anyone says, and that it's still written as such tells us that when they did the last orthographical reforms, it was still be considered a diphthong.
(And this is where I obnoxiously, pointlessly interject why I am so happy I learned Scottish Gaelic before Irish because no Scottish Gaelic speaker muddies that diphthong in words like this. Sorry, just had to do that. And yes, I know, it's obnoxious.)
Anyhow, because this diphthong ends with slender vowel, the following vowel sound in airgead must conform to the rule.
A couple finer points here: the r is slenderized here too (but it's often hard to hear--it's a breathier, lighter r sound that does not occur in English phonology. You can hear it more clearly in the recordings for air), however r is not followed by a h because h will not be placed after slenderized liquid consonants (r, l and n). And yes, it sounds like airgead is three syllables--that is the intrusive vowel that happens when r comes right before another consonant that r doesn't blend with. So airgead end up sounding more like air(uh)gead.
Oh my! Thank you sooooo much for such a detailed explanation! I actualy stopped my studies for a while these days and then when I come back I have this :) I understand it now, go raibh míle maith agat!
Ps.: I was also thinking about which one would be better to study first, since I'm planning to learn both :D
Remember that the primary form of any language is the spoken one and the written one is secondary. A word is in the first place a sequence of sounds, and only afterwards (if at all) of letters. This word happens to contain two slender consonants between broad vowels, and the way to write this in Irish is by inserting a slender vowel letter on either side. Writing argad would make it look as though the consonants are broad too, which they aren't.
The only examples that I can find of "gach am" in the first few pages of an Internet search are either in Scottish Gaelic, obvious machine translations (dodgy websites with product descriptions in "Irish"), or very archaic documents from the 16th and 17th centuries.
I don't think that you can swap am for uair in most cases.
It is about "simple presence" vs. "continuous presence". http://www.englishpage.com/verbpage/presentcontinuous.html It seems in Gaeilge the continuous presence would need "ag" + basic shape of verb: "Táim ag rith" = "I am running".