‘Half Irish’ doesn’t mean anything, and birth certificates don’t show nationality. Irish citizenship law says that anyone born in the island of Ireland up to a fairly recent date (2001, I think) is an Irish citizen. The children of those are also Irish citizens wherever born. It’s not a matter of ‘applying’ for citizenship in that situation, but I can see that some people would like to apply for an Irish passport to prove citizenship. And both British and Irish law allow dual nationality and allow you to hold both passports at the same time (some countries, including, in the past, Ireland specifically prohibited dual passports).
A citizen born outside Ireland doesn’t automatically confer citizenship on his/her offspring, but they can apply for entry on the Foreign Births Registry, which involves giving evidence of Irish descent. It’s fairly easy to do, and I recall the cost for my son was fairly modest (c£35 about 10 years ago) but be aware that the DfE does make telephone calls to witnesses for registration and passport applications, and does reject the application if they can’t get hold of your witness in a reasonable time.