I know the phrase
我が家 means "our home" or "our family". My question is when would you use it, as opposed to say
私の家族? It seems poetic to me, or something that wouldn't exactly be used in everyday conversation.
That you are proud of your home (family) or that you have a deep affection toward your home (family). You could think of it partly as meaning "my dear home" or "my beloved home".
Not used in everyday conversation (used more often in the written form) and shows some form of rigidness and politeness. (As Tsuyoshi Ito mentions in the comments, 我が家 is probably more common in everyday conversation than other forms of 我がX, for example 我が家へようこそ would be common I think).
The above also applies to 我が母校、我が国、我が家族、我が妻、我が故郷、etc. They all imply that you have a deep relationship, you are proud of them, show affection, etc. I often hear 我が国 when the prime minister of Japan gives a speech (perhaps the other ones might be used in a speech also). I also feel that it would be more often used by men, but that might be my own personal experience only.
There also is the form 我が子, however, this form doesn't necessarily refer to the person speaking (I feel like there are other examples of this, but I cannot think of any at the moment).