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First you need to understand the difference between a dominant and a recessive gene. Dominant genes manifest if they're present, recessive genes manifest only if there are no dominant genes present. A quick example is blood types, 0 is recessive, both A and B are dominant, you have two genes that specify your blood type, if both of them are 0 you are type 0, any other combination with 0 you're not 0, i.e. 0A or A0 are A, while 0B or B0 are B. This means that a person with blood AB can't have a son with blood 0, because his son will either have one A or one B inherited from that person.
Genetic diseases that happen because dominant genes are hard to miss, if you have the gene you have the disease, however genetic diseases that need recessive genes can be carried for generations without anyone manifesting symptoms. But of two persons have the same recessive gene it's quite possible that their children will have both of the genes be that one and manifest the illness. The chances of two random people having the same recessive genes are quite slim, but the closer people are genetically the higher the chances that they have the same recessive genes. Using blood type as an example, if a parent is AB and the other is B0 their children have 0% chance of being 00, but they have a 25% chance of being A0 and 25% chance of being B0 (the other 25% being AB and BB). Now if their children A0 and B0 have a child of their own that child has a 25% chance of being born 00, whereas if any of them had a kid with a AB, AA or BB the chances would be 0%.
Does this mean that blood type O will become rarer over time and eventually disappear? Does it also imply that someone, like me, who is O- has a family tree with fewer branches on it than most people?
It is possible, but genetics is far more complicated than this. There are also the epigenetic factors for gene expression that is a field of ongoing research. Nothing is ever actually as simple as the example but it is a good example for illustration of the concept.
Recessive alleles like the O blood type are preserved when paired with a dominant allele. So parents that are AO or BO can have children that are OO. The recessive allele's effects are suppressed, but it doesn't disappear. It keeps popping back up in future generations. That was one of Mendel's key discoveries.
The frequency of alleles circulating in a population is affected by drift and selection. Assuming no or very weak selection against type O, it's a matter of chance each generation if there are fewer or more children with type O alleles. The O allele could drift to 100% (also called being fixed) or to 0% by chance. This takes a very long time when the effective population is large but is faster for small, isolated populations. There are some variant alleles that are circulating in humans which have been there since before our split with chimps and gorillas.
The largely mathematical field that studies this is called population genetics.