The structure of the international red cross movement is a bit confusing. There are completely independent national societies in most countries (E.g. The American Red Cross), as well international organisations based in Geneva (The ICRC and the IFCR).

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.