A single random change in the chemical make-up of a DNA molecule will naturally be passed on to the next generation of cells produced. Afterall, one of the steps in the process of cell reproduction is the splitting of the original DNA molecule, whereby each half itself becomes the precursor to a brand new replicate DNA molecule. Each half is like a "photo negative" which will compliment a corresponding sequence (the other half).
The information encoded in the pattern of the original DNA is communicated to the next generation of DNA molecules which dictate the structure of resulting cells. Many would probably agree that at least one single change could happen at least one time to a least one strand of DNA in the life of a single organism. Considering the constant relentless bombardment from external sources (environmental factors such as weather, climate, radiation, local competition, etc.) experienced by all creatures, it wouldn't be too far fetched to assume that accidental mutations are not all too infrequent. One could also assume that these mutations would have been influenced by the local environment, casuing affected organisms to have noticable genetic responses to accomdate their surrounding environment.
Now given a constant propensity to undergo change, and given that each change is communicated via "molecular blueprints" (i.e. DNA) to the next receiving generation, it would logically seem inevitable that many changes would accumulate over many years. A single mutation per lifetime was used in the above paragraph as a conservative statement to build my argument. I would contend that many more changes happen to the DNA within organisms than just one. DNA is replicated many times in just a single organism throughout its lifetime, and each time can mean another iteration in generating another DNA pattern. The gradual accumulation of many such changes is the mechanism driving evolution, which doesn't just occur in bio/eco systems.
Look at law, the economy, technology, knowledge...
Just think how the gradual accumulation of many ideas, discoveries, inventions, improvements on those inventions, laws, etc., have caused huge impacts on the human way of life. Many small insignificant changes coupled together can have an overwhelmingly large impact. Evolution is a natural consequence of passing information on to recipients. Whether discoveries or organisms, over the years more and more information will accumulate through trial and error, what works and what doesn't. In nature, what works is those that survive and pass on "what works" to their offspring. And what doesn't work, dies, and the passing on of those traits naturaly ends right there.
Just think about this, how many of you look EXACTLY like your parents trait for trait? How many generations do you think it would take for the descendants of one bloodline to no longer resemble the original parents? Do you think given billions of years for changes to accumulate via this process, you couldn't have a rich trait diversity amongst descendant offspring. Do you even stop to think about how long a BILLION years really is? Could at least a trillion small insignificant changes have a large impact?
Imagine we played the game of telephone and a TRILLION (or even a QUADRILLION) small changes were made to the original sentence...