1.
vessel, container, dish
A container or dish used for holding food, drink, or other items. Often refers to pottery, ceramics, or tableware with aesthetic value.
2.
capacity, caliber, character
A person's capacity, caliber, or suitability for a role. Used figuratively to describe whether someone has the personal qualities needed for a position or responsibility.
In sense 1, 器 is closely associated with Japanese food culture and aesthetics — the presentation of food on carefully chosen 器 is considered an important part of the dining experience.
In sense 2, 器が大きい (big vessel) means someone is generous and broad-minded, while 器が小さい (small vessel) means someone is petty or narrow-minded.