Influenza is one of the most pesky and occasionally frightening of common human viruses: it is more harmful than the common cold, yet mutates and combines to form new viruses very quickly, so that one cannot simply escape it through a long-term vaccine like those now available for former killers like smallpox, measles, and polio. Flu viruses combine to form deadlier viruses in much the same way that all other viruses alter their makeup: mutations in copies of the virus are replicated in an infected person, and then, especially if those mutations make the virus more contagious, they begin to spread throughout the population.
In general, the human immune system is quite effective at detecting foreign threats within the body and producing antibodies to mark them for destruction. It is even more effective at detecting these threats the second time they enter the body. For this reason, once you have been properly vaccinated or after you have already suffered from a particular viral illness, you will usually never get it again. The immune system will immediately recognize the virus after you are exposed, and eradicate it before you begin to suffer illness.
Mutating and combining to form new viruses are the virus's strategy of defeating this strength of the immune system. The principles of viral mutation and evolution are essentially similar to those of evolution of higher life forms, except on a much shorter and more dramatic time scale: viral RNA is simpler, and viral replication occurs very quickly and in very large numbers (a single infected person will have millions of copies of the virus that has infected them in their bloodstream, for instance). When the virus has changed enough that the immune system no longer recognizes it from past experience, it can infect you again. For this reason, viruses which mutate dramatically and rapidly, like the common cold and influenza, can seem to infect us again and again. In fact we are being infected by slightly different viruses each time.
This mutation process, however, only explains why flu viruses can combine to form new viruses capable of infecting us again - it does not explain why the flu has such a nasty habit of combining to form deadlier viruses. Global influenza pandemics such as the H1N1 or "swine flu" pandemic of 2009 occur every several decades, and occasionally they can be devastating indeed: the 1918-1919 Spanish flu pandemic swept around the globe in a matter of months in an age before air travel, and killed tens of millions of people, more than the entire First World War (which, at the time, had just ground to an end). We were fortunate that the swine flu pandemic turned out to be only moderately more worrying than the average annual flu season. However, there remain fears that the next major flu pandemic will come from an H5N1 strain of flu currently common to chickens (and referred to as avian flu).
This last bit hints at what makes the flu so prone to combining to form deadlier viruses: it is a disease that does not just affect human beings. Influenza actually has several natural species reservoirs: humans, but also pigs (swine flu), birds (avian flu), and less commonly horses (equine flu). Swine, avian, and human flu are capable of combining together to form deadlier viruses. In most circumstances, this does not happen: most strains of avian flu only affect birds, most strains of human flu affect only humans, and so on. Occasionally, however, a mutation occurs which allows the virus to jump the species barrier - so that, for example, a strain of avian flu begins infecting humans. This is precisely the vector public health officials fear could lead to an H5N1 avian flu pandemic.
This species-jumping capability is what makes the flu so dangerous. On those uncommon occasions when it does so, it brings with it a whole host of mutations from its time in the other species, mutations against which no human immune system has had time to prepare a defence and which can make flu viruses particularly deadly. Over time, the new flu strain will usually combine with existing human flu viruses and form less deadly strains. However, in the meantime, as occurred in 1918-1919, tens of thousands, hundreds of thousands, or even millions of people can lose their lives.
In the swine flu pandemic of 2009-2010, for example, the virus we are confronting has actually crossed the species barrier multiple times, so that medical scientists have detected genetic material in the H1N1 pandemic virus which originated in older human and avian as well as swine flu strains. The reason it was originally known as "swine flu" was because the last of these jumps was from pigs back to humans, somewhere in Mexico in spring 2009.