Conventional web hosting operates under the assumption that the hosting provider will comply with legal requests to remove content or terminate accounts. When a major entertainment company sends a DMCA notice to AWS, Amazon removes the content. When law enforcement requests account information from a US-based provider, the provider typically complies. This responsiveness to external pressure is how mainstream hosting works.
Bulletproof hosting inverts this assumption. A bulletproof host operates in a jurisdiction where foreign legal instruments have limited or no binding force, maintains explicit policies about which requests they will and will not comply with, and builds operational independence from parties that might apply pressure. The result is hosting that stays up when mainstream providers would shut it down.
The term 'bulletproof' is a colloquialism, not a legal or technical standard. Different providers have different thresholds for what they ignore and what they act on. Understanding a specific provider's actual policies is more useful than the generic label.