An ultimatum is a "do this, or else" sort of a statement that one country makes to another. In this case, Austria-Hungary was telling Serbia that Austria would go to war unless Serbia did a number of things. Austria purposely made the terms of the ultimatum very harsh because they wanted it rejected -- they wanted war.
Among the things they demanded from Serbia were:
- Serbia's government should renounce all propaganda against Austria -- propaganda about Slavic nationalism.
- Serbia had to arrest any Serbian officials involved in the assassination of Archduke Franz Ferdinand.
- Serbia had to allow Austrians to come into Serbia and conduct the investigation -- to allow Austrians to come in and decide who should be arrested in connection with the assassination.
This last clause was the one that was most unacceptable because it is very unusual for a country (Country A) to let a hostile country (Country B) come in and conduct police operations inside Country A. It's like letting Country B take control of Country A and that's not acceptable.