Actually, changes in the sequence of the bases in DNA happens in our cells all the time, as the DNA in each cell is very long and easily damaged. The amount of DNA in each of our cells is staggering.
Each one of our cells contains DNA that is just over 3 billion letters long (that is 3 000 000 000). This DNA would stretch two meters long if you took it out of the cell. Consider that the complete works of Shakespeare and the Bible each contain around just 3 million letters! Every time a new cell is made, those 3 billion DNA letters need to be copied to make a new set.
Given the number of letters, mistakes are always made leading to the sequence becoming changed. This is called a mutation. Most of the time these changes are not noticed, and the function of important genes is not changed. If a mutation occurs that changes the way a gene works, then the behaviour of the cell can change for example a cell can begin to grow at the wrong times. A mutated gene that causes cancer is called an oncogene.
A single mutated gene does not by itself cause cancer. Cells accumulate multiple mutations over time. When a cell has several mutations in important genes that they become cancerous. Throughout our lives DNA in our cells is getting damaged and mutated. If the accumulation of these changes in the DNA of certain cells happens to be in certain genes, then that cell may become a cancer cell.