A hybrid is something that is a combination of two normally different things. So, for example, a hybrid car is a car with an engine that burns gasoline like a normal gasoline-burning car, but in the process the hybrid car also charges a super-powerful battery that it can run on when the engine is turned off. This allows the car to be more fuel-efficient.

A hybrid organism is a hybrid between two types. These can be subspecies, like the breeds of dog, or sometimes entire species, such as a mule being a hybrid between a horse and a donkey. Hybrids are created by interbreeding; a mule has one parent who is a horse, the other who is a donkey, and so being the child of two different species, the mule has aspects of both.

Plants do this too: for example, a pluot is a hybrid between a plum and an apricot. The pollen came from a tree of one species, but the ovule (plant egg, basically) came from another.

Purebreds refer specifically to breeds, where an individual is a member of a specific breed of what-have-you. Nectarines are a breed of peach. A hybrid between the more common subspecies of peach (which has hairy skin) and a nectarine would not be purebread.

Non-disjunction does not ensure that there will be differences in traits between generations. Non-disjunction is a mishap whereby the chromosomes do not split during cell division that creates spores (plants) or gametes (animals). When non-disjunction happens, it usually causes problems in the offspring . For example, Down Syndrome in humans is caused by non-disjunction of chromosome #21 , resulting in a child having three copies of the chromosome in each cell, instead of the usual two copies. Children who afflicted by this are sterile, suffer mental retardation, and other problems.

What you are probably thinking of is segregation. Segregation (in biology, not politics) refers to the fact that alleles do not blend into each-other, so even if you are not expressing a genetic trait, you can still pass it to your offspring.

A purebred means that both parents of an individual come from the same species and variety/breed as the individual. A hybrid would mean the parents of an individual do not come from the same variety of the same species. We often hear this term used for dogs - so Labradors and poodles would both be called purebreds, but a Labradoodle would be a hybrid. Non-disjunction is a failure of homologous chromosomes or sister chromatids to separate appropriately during cell division (either meiosis or mitosis).

Non-disjunction can result in sister chromatids being segregated into a single daughter cell instead of two daughter cells, so the daughter cell that loses a single chromosome is called a monosomy (45 chromosomes in human cells) and the daughter cell that gains a single chromosome is called a trisomy (47 chromosomes in human cells). These two daughter cells would both stand in contrast to their parent cell, which was assumed to have no imbalance in the number of chromosomes (46 chromosomes).

The monosomic daughter cell has now lost the function of all genes on the chromosome it did not receive, but the trisomic daughter cell does not necessarily gain the function of all the genes on the chromosome it did receive. In this sense, the daughter cells will have different traits from the parent cell, but these differences show up as syndromes and disorders and not as beneficial effects, so the phrase "differences in traits" may be misleading.