Mostly, people are talking about competition between sperm produced by different males. In humans, this means women that are into speed dating.
The non-paternity rate is an upper limit to the rate of sperm competition: in many, probably most cases of non-paternity, the woman has not had sex with two different men in a short period of time. But the non-paternity rate is low!
So classical sperm competition is insignificant in humans, at least in every population we have any data for. There are those that argue that back in the stone age, things were far more crazy [hunter gatherers were highly promiscuous, they say], but they’re utterly full of shit.
Now if you go far back enough, a few million years, things were different. Chimps are highly promiscuous, and have specific adaptations for sperm competition, for example a protein that causes ejaculate to form a plug. Humans have a non-working version of that protein, which shows that sperm competition used to matter in our ancient ancestors, but hasn’t for a long time.
Still, a number of genes involved with spermatogenesis evolve quite rapidly. We know several ways in which selection might favor a lot of change. One is that sperm cells from a given male compete with each other. The other is sexual conflict: the sperm and the egg have conflicting interests. An allele that increased a sperm’s chance of being first to the egg would be favored. On the other hand, it is absolutely vital that egg not be fertilized by more than one sperm, so barriers to over-fertilization must exist. Sperm adapt to vault those barriers more effectively, eggs evolve better barriers, etc. It’s a Red Queen situation.
A lot of ink has been spilled about all the special physical and psychological adaptations in humans driven by sperm competition of the first kind. They don’t exist.
Baker and Bellis wrote a lot of nonsense along these lines back in the 90s, sprinkled with fascinating false facts like 20% rates of non-paternity in Britain. Loons, both of them.