The usual bad habit in school Irish is to use clann, which is completely, earthshatteringly, capitally punishably wrong. (OK, it is true that it has started to infiltrate the speech of native speakers too, pitiably enough.) Let’s have a look at this.
Clann is wrong because it basically means the children or descendants of someone (an bhfuil clann agat? means “do you have children?”), not the family in the modern nuclear family sense. In fact, the raw English loanword family (possibly it could be written feaimilí) has some currency in relatively traditional speech (in folklore texts, for instance).
The “standard” word is teaghlach. It is often suggested that this is a revived literary word from Early Modern Irish, but I definitely disagree. Séamus Ó Grianna makes use of it, although I was left with the impression that it means “children, descendants” in his Irish. However, when I mentioned this to more learned Irish speakers, they had no problem providing quotes from native literature showing that teaghlach has traditionally been used in the sense of “family”, so that it is not wrong or unnatural to use it for modern nuclear family. (I do tend to think though that it was originally preferred by language planners due to the fact that it sounds a little similar to the Welsh word teulu.)
Then there are the words muirín and muirear. These are used when we speak of the family as a burden, as people to be clothed and provided with food and shelter. Family planning is pleanáil na muiríne in Irish – note by the way that muirín is feminine.
Comhluadar has basically the meaning “(jolly) company” and consequently “fun” at least in some dialects (a similar development has affected the synonymous cuideachta in Munster, where it has the form cuileachta, c’leachta), but in Connemara Irish, comhluadar means “family”. Comhluadar would otherwise be a very good word for “family”, but in this sense it is a provincialism, and might not be understood by non-Connacht speakers of Irish.
Conlán was the word for “family” in the defunct dialects of East Ulster. I guess it is just fine to use it so, if you are making a conscious attempt to imitate them. (Examples of conlán in this sense can be found in Sgéalta Mhuintir Luinigh, a recent book of folklore in Tyrone Irish.)
Líon tí means family in the sense of the people of a household. Líon means full number and tí is the genitive of teach, house, so it’s basically everybody living in the house.
Cúram means “care” and thus “the people I am responsible for”, i.e. my family, but rather in the sense of clann, i.e. children.
Fine is used in the figurative sense, i.e. a language family, for instance.
PS: And, as Mike pointed out in the comments, there of course is the word muintir, which is basically translated with “people”, but often means “family”. Muintir na hÉireann means “the people of Ireland”, but mo mhuintir, “my people”, i.e. basically the people associated with me, usually means, “my family” (it can also mean “my parents”).