ConvenienceConvenience is the quality of ease or reduction of effort that something allows. If something is a convenience, it allows you to complete a task more easily, or to complete it with less effort.
Ex: Since his car is being worked on this Saturday, the rescheduling of Saturday's event actually proved a great convenience to Bruce. He can now attend the event on Sunday and not have to make other transportation arrangements.
Ex: Having a microwave in the kitchen is a great convenience, even though it takes up space. It can warm sausages, melt butter, and cut the cooking time for potatoes by more than half.
The expression at someone's convenience means that something is scheduled for a time that best fits the person's schedule or needs, as in we can make the appointment at your convenience. If you use the expression for someone's convenience, you are saying that something is designed to make a task or situation easier for them or to reduce their workload, as in for your convenience, we left a map of town marked with all the local sights in the guest room.
Ex: Let's have lunch and catch up soon. We can schedule a time at your convenience since my schedule is very open right now.
Ex: We hope that your visit to our campground is enjoyable. For your convenience, showers and restroom facilities are located within a half mile of each campsite.