In either case- you as a recipient will not owe any tax on this gift. You will not include this as income or report it on any form. You are also not required to file and report the foreign gift unless it is $100,000 or more.
Regarding your mother giving this gift - she can give upto $12000 in a year without owing any gift tax or giling a gift tax return. She can, however, give gift over $12,000 and not owe any gift tax but she will be required to file a gift tax return. (see detailed description below).
If you get a gift from a foreign person than he is not subject to gift tax laws in US but he may be subject to gift tax laws of his country of residence.
Normally a person can give up to the annual exclusion amount $12000 for 2008 to a person, every year ($ 24000 in 2008 if spouses(US resident) joins in for the gift) without facing any gift taxes and note that such amounts do not count as part of your $1,000,000 lifetime total.
Further, IRS allows a person to give up to $1,000,000 in gifts, total, in their lifetime, before they start owing the gift tax. (This gift is not per (donee)person but its a per donor limit). So you can make gifts that are worth up to a million bucks during your lifetime without paying the gift tax. . Even if you do not owe a gift tax because you have not reached the $1,000,000 limit, you are still required to file gift tax return if you made a gift that does not qualify as excludable.
Let me know if you have any question.
Please note: This advice is provided with the understanding that all the relevant facts have been provided by you. Any change in facts might affect the advice given and hence may not be relied on in such cases. Nothing contained in this reply was intended or written to be used, can be used by any taxpayer, or may be relied upon or used by any taxpayer for the purposes of avoiding penalties that may be imposed on the taxpayer under the Internal Revenue Code of 1986, as amended.
Yes, you can claim it that way. The payment does not have to come from them individually. They can elect that they made a joint gift.
Btw, I was not aware of your follow up question else I would have replied.