What is the relationship between sexual and and natural selection?

To Darwin's point of view, sexual and natural selection are two different mechanisms. Then, we tended to consider that sexual selection is part of natural selection. The part of natural selection that is not attributed to sexual selection is sometimes called ecological selection. Today, our understanding of these mechanisms brought us to consider the meaning of these words slightly differently.

Natural selection is not anymore the result of a struggle for survival. It is any process yielding to a change in allele frequency through a long-enough term difference in fitness. Fitness is a index of how much an individual, a gene or whatever, leave copies of itself after a long and short enough amount of time (one might not agree with this definition I guess). Take two strains of bacteria in two petri dish, consider they are one population and let the bacteria grow. One strain will grow faster and the allele frequency will vary over time. It is the result of natural selection although there were no competition, no struggle at all given that they were not in contact.

Sexual selection is a type of natural selection that involves sexual competition. There are two types of sexual competition, the so-called inter- and intra-sexual competition. Inter-sexual competition means that one sex chose the other one, while for intra-sexual there is no choice from the other sex. For example, if you see two animals fighting, they either fight to have the physical acces to the other sex (intra-) or they fight and then, let the female decide which one is the best to mate with (inter-). These two concept of intra- and inter-sexual competition might somehow overlap in some specific case. I would argue that the concept of sexual competition and therefore of sexual selection have never been accurately defined. Therefore, I would not complain if one prefers to give up the concept of sexual selection to use only the concept of natural selection.

Many articles state that sexual and natural selection act on a trait with the same or opposite directions. Given the above argumentations, you could as well say that there are two different forces of natural selection. if the trait is at state 'A', then the individual is very attractive to the other sex but is likely to not be able to escape a predator. This is the case of most of the trait we think of when talking about trait evolving through sexual selection. See Zahavi's handicap principle for more information.