In
polyandrous species, even males who are mating successfully may have trouble siring offspring. For
Mr. Right no. 2, it's a simple numbers game whether his sperm, the last guy's, or the next guy's will be the lucky choice of the female's
ovum.
One oft-seen solution to this is territorialism: guard your mate jealously, and make it imprudent for her or her potential suitors to make a connection. But for many species, this method only goes so far: constantly fighting over one or more mates can have a huge energy cost, and may be unsuccessful -- especially since she may have a say in the matter.
Sperm competition changes the field of competition to the inside of the womb. Males attempt to produce more, better sperm than their peers, increasing the success of their matings rather than preventing their opponents'. Each extra sperm in their ejaculation is another ticket in the big raffle of Life.
One can clearly observe this strategy as an energy investment in the size of the testes. The classic example is a comparison between gorillas and chimpanzees. The social structure in which gorillas live supports the territorial strategy, allowing a successful male to maintain tiny testes (relative to his bodyweight). He's the only one playing the lottery! By contrast, chimpanzee males must have larger testicles to produce more sperm, since they live in more dynamic social groups that support multiple mates for each female. In fact, you can often infer things about current or past social structures in a species based on the size of the testicles.
Sperm competition usually refers to quantity of sperm, described above, but there are other interesting adaptations out there. Some species use hook sperm -- deliberately crippled sperm capable of waylaying other sperm -- to sabotage their competitors' exertions post-coitus. In fruit flies, the male employs a toxic sperm that may discourage the female from subsequent matings. And on a more positive note, mice sperm (from a single male) cooperate using formation flying to speed their approach to destiny.
Sources:
http://www.anthro.ucdavis.edu/features/stp/stpballs.htm
http://www.nature.com/nsu/020708/020708-10.html