"An Teachta Dála."
It's also worth noting that in English, they're generally referred to as "TDs"
Like what we call an MP* here in Canada (and other places)?
*Member of Parliament
The title "Member of the Legislative Assembly", or MLA, is given to someone election to the Northern Ireland Parliament. The Irish is Comhalta den Tionól Reachtach (I had to look that up - I've never heard it used, though I imagine it must be used sometimes).
A TD is not an MLA, and an MLA is not a TD. While you could argue that a TD is a member of a body that can be described as a "legislative assembly", it's definitely misleading to refer to it as "the legislative assembly", as there is a body called "The Legislative Assembly" in Ireland, and it's not the Dáil.
I think the closest-to-home (for me) equivalent would be an assemblyman/-woman of the California state assembly. Right?
According to wikipedia, "The California State Assembly is the lower house of the California State Legislature", so yes, An Teachta Dála would equate to "assemblyman/-woman".
But it should be noted that, even though Seanad Éireann is referred to as "the upper House", TDs don't aspire to become Senators in Ireland - membership of the Seanad is seen as a fall-back position for failed or aspiring TDs!
It seems an odd omission. While "Deputy" is used one it's own as a title (Deputy Kelly, Deputy Ryan, etc), the normal reference in Ireland would be to "the TD", rather than "the Deputy", and I'm pretty sure that I've heard RTÉ refer to Dáil Deputies in news reports.
The Oireachtas website says "A Member's official Irish title is "Teachta Dála" which in English means "Deputy to the Dáil"; Members are generally called "TDs" or "Deputies"" which I suppose gives some precedent, but I still think "The Dáil Deputy" would be a better translation than just "The Deputy", but as long as Duolingo accepts "The TD", I'll live with it!
Deputy implies substituting for someone. So who do they substitute for? Like the deputy sheriff substitutes for the sheriff???
They are substituting for the people who elected them. Just as a Sheriff's Deputy is authorized to act on the Sheriff's behalf, a Dáil Deputy is authorized to act on his constituents behalf in the Dáil.