I am thinking of specific examples. In much the same way, David Corfield mentioned groupoids.

I am personally not a big fan of the general theory of loops. In part, my own disinterest is because I have not found an application. On the other hand, I have seen enough to believe that Moufang loops are interesting even if I personally don't know a lot about them. Still I like the idea of algebraists thinking about the structures of loops because they find them interesting.

Closer to my own interest is the idea of quandles. These were introduced essentially in the 1940s, then again in the late 1970s and early 1980s, rediscovered, and have only found some greater applicability because quandle cohomology gives interesting topological invariants. The idea, apparently was natural: it was discovered, forgotten, rediscovered, forgotten, and found to be applicable. Nevertheless, some of you might find it to be be a fringe notion. Even knot theorist might believe that there is not much in the quandle concept because the information in the quandle is present in the fundamental group and a peripheral subgroup.

I think Tim's articulation of Ronnie's list should include that the algebraic concept yields a more concise language in which ideas can be expressed.