If it weren't for sexual selection, evolution itself would be a passive process. Natural selection isn't sufficient in the evolution of life, because it does not deal with predictions of future selection pressures. For example, if a massive natural disaster were to wipe out every single collared pigeon on Earth, then there would be no process in place to bring the collared pigeon back to life. Since the development of the pigeon from its ancestor must have taken a very long period of time, it is a really inefficient idea to just let the pigeon's fate hang by chance. Admittedly, if all members of that species were to be wiped out, the chances of it being reintroduced would be very close to zero.
So, how does sexual selection work to avoid such fates of death of a species? Firstly, let's establish that sexual selection is not the process of reproduction, or self-propagation, but the process by which certain properties are chosen over others to deal with potential future selection pressures. Of course, the most basic form of choosing those properties is reproduction itself, especially in some birds and mammals where courtship behaviour is an integral part of their life.
Think the peacock, perhaps an animal frequently used to explain evolution. The peacock's tail is brilliantly coloured. Certainly, this cannot be an adaptation which improves survival, because it makes the peacock a lot more likely to be predated, as it makes itself very obvious and noticeable. Hence, a different process must have led to the colourful tail to be selected for. This process, sexual selection, can be quite subtle and indirect. Let's assume that for a long time, there have been no significant natural selection pressures where peacocks lived. So populations of peacocks lived peacefully with no threat from the environment. Instinctively, the peacocks were wired to be prepared for a pressure to come. Of course, there is no way for them to have known what that pressure would be - change in temperature, lack of food, etc. Therefore, sexual selection acts on selecting properties which are generally suited for selection pressures. For example, regardless of selection pressure, intelligence would be a good property. Strength and agility are others.
Now, you may think, what has a brilliantly coloured tail have to do with either of those advantageous properties? Well, surviving despite being more prone to predation due to the coloured tail must exhibit superior qualities. Intelligence for being able to raise and lower the tail depending on situation (mating or standing by), strength for dealing with the unwanted predatory attention from predators as well as other competing peacocks, and ultimately, all the good qualities that the peacock has inherited from his own father who has obviously not only made it alive, but made it to mate as well.
These qualities are broadly targeted to future selection pressures. They do not necessarily accurately predict the selection pressures, but they are a way to deal with them, like a best bet.
Look around at the human species' best bets: some think the world might heat up, so they vote for the Green party. Some think humans might die out, so they may be homophobic. Some think the Earth might be destroyed by an asteroid, so they invest in space travel research.
But there are certain properties which have become universal over time, the properties that everyone bets on: intelligence, humour, friendship, love. Whatever the next selection pressure, we bet these properties will somehow save us.
Peacocks do their betting by reproduction. Humans reproduce, too, but they have many other ways of doing their betting and selecting the properties they think will come on top. Find out more about sexual selection in humans, and how intelligence has shaped, and is still shaping, our evolution in the next post.