Small correction - it's "cycles mod boundaries". In general, a quotient object R/I is read "R mod I" (top mod bottom). What's a cycle? It's something in the ke

Small correction - it's "cycles mod boundaries". In general, a quotient object R/I is read "R mod I" (top mod bottom). What's a cycle? It's something in the kernel of the boundary map. To be in the kernel means the boundary map takes it to 0. So heuristically, cycles are "things with no (zero) boundary". For the quotient cycles/boundaries to make sense - for any quotient to make sense - the bottom should be contained in the top. Like in Z/6Z, the integers mod 6, the "6Z" (the integers which are a multiple of 6) is contained in Z (the integers). In this case, for "cycles/boundaries" to make sense, a boundary should be a cycle - that is, a boundary should have no boundary. This is (again heuristically) what the definition of a boundary map (d^2 = 0) means. Consider your disk/circle analogy. The circle is a boundary (the boundary of the disk). Does the circle have a "boundary"? No - the boundary of a boundary is 0. When the quotient cycles/boundaries is nonzero, you have nontrivial homology. Think of nontrivial homology as "having a hole". To see an example of this, take the plane but remove the origin (0, 0). Consider your circle again (say the unit circle x^2 + y^2 = 1). It's still a cycle, because its boundary is 0 (no boundary). But it's not the boundary of "something" (where "something" means roughly "something nice" - it's hard to explain this better without getting technical). So the cycle represented by the circle represents a nontrivial homology class (in the homology of plane minus origin).