Worth re-iterating: The American Red Cross and the International Committee of the Red Cross are independent organizations, with very different reputations.
Care to expand on this? Edit: Based on the downvotes, I'm guessing I wasn't really clear what I was going for - I hadn't heard about the International Committee before, so I was looking for some more information about how they differ.
In the US, the phrase "Red Cross" often is used to refer to "American Red Cross", which is the national society who deliver blood services and local disaster response. They've come under some criticism for apparent lack of transparency around how donations are spent, and for allegedly having very high overheads.
The IFRC (International Federation) is an international organisation that facilitates cooperation between the independent national societies.
The ICRC (International Committee) is a large international organisation whose mandate derives from the Geneva Convention, and whose work is focussed on armed conflict. They work with war casualties, prisoners of war, refugees, civilian victims of all forms, on any and all sides of conflict. They also help coordinate disaster relief in lots of cases just because they have the organisational ability to do it. In my opinion they're among the most trustworthy organisations that you could give to.