I know the broad strokes of genetics and inheritance, of dominant and recessive traits, and how a pair of chromosomes sort of send a random representative between them from each pair of genes.
What would happen if that selection weren’t random? Suppose there were a type of gene that was persistent, that is, of the two genes selected, it would always be the one passed on to the offspring?
If I understand your question correctly, then that gene* would have to have some mechanism for messing with chromosome distribution. Say you have the “persistent” gene on your dad’s chromosome 21 and the non-persistent version on your mother’s chromosome 21. The “persistent” gene has to be passed on, so when you have kids, you must pass on your dad’s chromosome 21, violating Mendel’s rules of independent assortment.
Alternatively, the mechanism could allow both chromosome 21s to be passed on, in which case your offspring has 3 copies of chromosome 21 (Trisomy 21, aka Down’s Syndrome - IIRC, a genetic disorder).
If the chromosomes didn’t randomly assort in meiosis - then a lot more of your offspring would look like each other and there would be less genetic diversity.
*From the context, I’m guessing you actually mean allele, because the simple answer would be that a gene is always passed on regardless of which chromosome carries it as long as that chromosome is paired (has a homologous chromosome.) The different versions of the gene on each chromosome are called alleles.
Actually, you mean allele, not gene, but those terms are commonly mixed up.
If there was on allele that always got passed on, then it would always get passed on. What do you mean “what would happen”? You defined what would happen. It would reduce the genetic variablility of the species and, over time, would either disappear completely (if it had adverse affects) or be present in all members of the species.
If you’re asking if there is such an allele, the answer is “no”.
Such a hypothetical gene could also act by overwriting the other gene of the pair in an organism. That is to say, I inherit a purple-skin gene from Mom and a green-skin gene from Dad, but the green gene in all of my cells somehow acts to replace the purple one, so I now have two green-skin genes. One of them will be picked at random for each of my kids, but it doesn’t matter which one, since they’re now identical, and my kids will also end up homozygously green.
You could do it with a virus that copies itself only into a specific segment of a specific pair of chromosomes. It’s not impossible, but I don’t think anyone’s found anything like that yet. Of course, some would then argue that the virus was not part of the human genome and we’d end up in semantic purgatory for years.
I suppose I meant, “What would happen to the population over time,” which you did answer.
Thanks, everyone. 
I’m just speculating on genetics for the purposes of a sci-fi story idea I had, but this probably won’t get the result I’m looking for.
I think you mean “yes”.
I once worked on a mutation that seemed to enhance sperm carring the allele (a “super-sperm” phenotype). It therefore seemed to be transmitted at greater than average numbers to progeny as sperm carring the allele seemed to fertilize eggs more efficiently than wild-type. Granted the progeny mice weren’t as healthy and didn’t reproduce as well, but there were more carrier and affected progeny mice in a heterozygous breeding cross than wild-type (the 1:2:1 ratio was off by a bit).
That’s all that will happen in a situation like the OP. Over generations, such an allele will act like it has positive selection. Remember, when Darwin came up with natural selection and when Willhelm Johannsen, Morgan, Muller, Bridges, Dobzhansky, and the other early geneticists started attributing things to “genes” nobody knew about DNA, base change, mutation, etc. So it doesn’t matter what is happening molecularly, at least on a population level kind of like the OP is asking. Phenomena like this do happen occasionally, especially if a mutation affects the cis wild-type copy like Chronos mentioned.
Here’s a bit more detail on segregation distortion, aka meiotic drive, from the same site. The phenomenon occurs but is rear.
If ideas for a science fiction story are what you’re after, maybe Wolbachia, a bacteria that manipulates the sex ratios of insects for its own purposes, might give you some inspiration:
http://www.sciencenews.org/pages/sn_arch/11_16_96/bob1.htm
http://www.berkeley.edu/news/media/releases/2002/07/03_paras.html
No, I meant “no”. I assumed that in the clarification sentence of the OP (the last sentence) was teh operative question-- that we were being asks about alleles that are always passed on, not allele that are passed on with > 50% probability, but < 100% probability.