Trust or "Amanah" refers to when you are given an object or are given responsibilities to take care of.
Taking care of an object would be to keep it safe and return it to the owner as it is when he wants it back. Taking care of responsibilities would be to fulfill them according to how they should be fulfilled.
The Prophet (SAW) even before prophethood was known as "Ameen" meaning "trusted one." This is because he was so trustworthy that people would give him their items to keep safe. The disbelievers would trust him with their objects even after he started preaching Islam.
A person lends you a book. Fulfillment of the trust would be to return the book as it is without damage.
You are employed for a job. Fulfillment of the trust would be to not waste work time or cheat your employer in any way.
A person is going away for some time and asks you to look after their house. Fulfillment of the trust would be to look after their house correctly.
Allah says in the Quran:
Indeed, Allah commands you to render trusts to whom they are due and when you judge between people to judge with justice. Excellent is that which Allah instructs you. Indeed, Allah is ever Hearing and Seeing. (4:58)
O you who have believed, do not betray Allah and the Messenger or betray your trusts while you know [the consequence]. (8:27)