A grant given by a university or foundation to a scholar for research or study is a fellowship. If you get a fellowship to do research on insects, it might bug your colleagues who didn't get one.
Use fellowship to refer to someone's company or companionship. Your grandmother might prefer the fellowship of people her own age, since they remember the same historic events as she does and know the same songs. A fellowship is also a community of people who share common beliefs or interests. A fellowship of knitters might meet weekly at a cafe in your town to exchange ideas and knit together.