Chromosomes can break! When they break, they get breakpoints. Ok, more seriously:
As a cell divides, during metaphase, the chromosomes all line up in the center of the cell. Microtubules attach to the chromosomes and pull them apart, so half the DNA ends up in each daughter cell. Before the DNA gets pulled apart, the chromosomes are free to recombine, so your chromosome 5, for example, is actually a mix of chromosome 5 from your mother and father. During recombination, the chromosomes must break and reattach. “Chromosomal breakpoints” refers to these places where they break. Occasionally something goes wrong and the reattachment happens in the wrong place…this can spell disaster. Usually the term “chromosomal breakpoints” is used in the context of some abnormality.
One of the early examples of this is the Philadelphia chromosome, a translocation between chromosome 9 and 22. This means a chunk of chromosome 9 ends up chromosome 22, and a chunk of chromosome 22 ends up on chromosome 9.
The term “breakpoint” refers to the position on the hybrid chromosome where the original location shifts, from 9 to 22 or vice versa.
There are other abnormalities besides translocation that also use the term “chromosomal breakpoint” though. For example, when comparing human and chimp, there are several “inversions” — where the DNA comes from the same chromosome, but a certain part of the chromosome is inverted. Such an inversion would also have breakpoints surrounding it. See this paper for an example.