What an amazing parable this is! I've been thinking about this for the last week or so and have been very struck by its profundity. Just a couple of remarks on its message...
1. Though it has a strong bent towards provision for the family of believers ("the least of these my brothers), I think its scope of application can hardly be limited to that (e.g. the "stranger" mentioned, Parable of the Good Samaritan, Gal 6:10). Even if we were to limit its scope just to believers we have absolutely more than enough poor Christians in the world to keep us busy for several centuries.
2. Care of the poor means tangible provision and support for practical and obvious need. This covers a wide range of things - food, relationship, acceptance.
3. Care for the poor is a necessary sign of personal and corporate regeneration. If we do not demonstrate care for the poor it calls into question whether or not we know Jesus. The whole point is that the goats don't recognize Jesus, they don't seem Him. They have a false Jesus in view and so are 'surprised' by the judgement. So, when you neglect the poor you neglect Jesus. To do nothing for them is actually to do nothing for Jesus. The real sin at work here is, then, not so much the sin against the poor, as the sin against Jesus. That is the deepest reason for why I am judged. As 1 Jn4:8 says - the man who does not love his brother, does not know God.