I am confused as I thought Mia Famiglia translated to My Family, but I was recently corrected and told it was La Mia Famiglia... what is the difference?
In English the definite article must not be put before the possessive adjectives, in Italian in general is the contrary:
Il mio vestito è blu correct
Mio vestito è blu wrong (if you say it we understand immediately that you are a foreigner).
However, there is an exception with singular names which represent family members, for example:
Mia cugina è logorroica correct
La mia cugina è logorroica wrong
Mio nonno faceva il muratore correct
Il mio nonno faceva il muratore wrong.
With famiglia the definite article is necessary:
La mia famiglia è di origini milanesi correct
Mia famiglia è di origini milanesi wrong.
As egreg correctly pointed out, with mamma and papà or babbo both forms are used, maybe with a slightly different meaning, for example:
La mia mamma è bella I say this because I love my mother and for me she's beautiful
Mia mamma è bella I say this as a matter of fact, she's a beautiful woman.
The definite article (or articolo determinativo in Italian) is used to reference a general category of people, animals, object or concepts.
In your case it is correct to translate My family with La mia famiglia (mind also the letter case) since also with the presence of the possessive adjective mia the use of the definite article la is always required.
You can find a very good reference in this Treccani Italian grammar