Some additions:
1. It does not always mean the same as broken. "Broken" like in "broken glass" would be "zerbrochen": "Zerbrochenes Glas".
So anything that ends up in many pieces when
destroyed is "zerbrochen" and things that just don't work anymore are "kaputt".
2. "Ich bin kaputt". You can increase the tiredness by saying "Ich bin total kaputt".
3. "Der ist doch kaputt!": He is weird, in a very
negative way.