To get into the nitty gritty (I know that some will appreciate it):
The 'e' is just there to 'soften' the 'b'. A soft 'b' is very similar to a 'hard' b, except that there's a little 'y' glide after it: almost like 'byan'. Other consonants have greater distinctions between their hard and soft variants, e.g. soft 'd' sounds more like the 'j' in jam, and soft 's' is like 'sh' in 'shout'.
If a consonant has an 'e' or an 'i' before OR after it, it's soft. If it has 'a', 'o', or 'u' before or after it, it's hard
So a lot of the time, pairs of vowels have two functions: one vowel is pronounced as a vowel, and the other 'flavours' the consonant it's sitting beside. You'll soon learn to distinguish these automatically. For the beginner, it's useful to know that long vowels (á é í ó ú) are always pronounced, so when a long vowel is beside a short vowel, the short vowel is just modifying the nearby consonant and is not said distinctly.
[I know, in the context of Irish,'hard' and 'soft' are usually called 'broad' and 'narrow' [leathan is caol], but I think the 'hard' and 'soft' terminology makes more sense to English-speakers and to anyone who's every tried to learn a Slavic lang]
Broad and slender actually, if we trust wikipedia here. But soft and hard are indeed easier to understand, though the Irish broad-slender pairs are not always similar to hard-soft pairs in a Slavic languages. E.g. Russian has two hard-soft pairs: s-s' and sh-sh', while Irish has one s-sh pair instead, which a Slavic language speaker would never consider as a hard-soft pair.
It sounds something like [bee-yan] said really fast, but it changes by dialects, and whether there's a shéimhú (that 'h' that seems to end up in just about every word!) There's great websites for learning Irish (www.gael-linn.ie, http://beo.ie/, http://focail.ie/Home.aspx), and http://www.focloir.ie/en/dictionary/ei/pronunciation translates and pronounces words in all the dialects so it should work). Hope that helps(:
I have a question about the audio: is anyone else hearing "bam"? Regardless of the vowel nuances, in the same way that I hear a very fricative [th] consonant when Spanish speakers pronounce words with a "d", I am hearing a slight "em" sound where the letter "n" sits.
I am solely an (American) English speaker and sometimes subtleties of language are lost on me, so I'm not sure if I'm imagining this.
I keep seeing people talk about the tips and notes at the top of the page but I don't see anything?
Just type lyrics "Mná na hÉireann" into your favourite search engine (note the reqired "h" after "na", though).
The Wikipedia article is probably your best bet, though:
The Irish for "woman" is bean.
"Initial mutations" such as lenition (an bhean) and eclipsis (ag an mbean) only occur when there is something to cause them. In the case of bean, it is a feminine noun, so it is lenited after the singular definite article an in the nominative case, but there is no definite article in this exercise, so there is no lenition.