
A trustless and distributed consensus system means that if you want to send and/or receive money from someone you don’t need to trust in third-party services.
When you use traditional methods of payment, you need to trust in a third party to set your transaction (e.g. Visa, Mastercard, PayPal, banks). They keep their own private register which stores transactions history and balances of each account.
The common example to better explain this behavior is the following: if Alice sent Bob $100, the trusted third-party service would debit Alice’s account and credit Bob’s one, so they both have to trust this third-party is to going do the right thing.
With bitcoin and a few other digital currencies, everyone has a copy of the ledger (blockchain), so no one has to trust in third parties, because anyone can directly verify the information written.