It is something that is good or useful for a person. For example, you can have a "matrimonio por conveniencia" (marriage of convenience), which means that you'll have benefits product of that marriage.
Another example could involve you and a friend talking about an oncoming Chemistry exam. After a few minutes you say: "Me convendría estudiar". This means that you should study, for you are not doing very well in that subject, and if you do (study) it would represent a benefit.
I hope this helps! :)