Competent cells are those that possess more easily altered cell walls that DNA can be passed through easily. These cells readily incorporate foreign DNA. On example of a competent cell is E. coli.
A cell that is not already competent can be made more competent through calcium chloride (a substance that creates organic ions) and heat shock. This is especially easy with cells undergoing very rapid growth.
This characteristic has an interesting potential application. With low levels of calcium chloride and heat shock that does not affect less-susceptible cells (in this case, those that do not have very rapid growth), it may be possible to make rapidly-dividing cancer cells competent enough to take up DNA that will replicate with the cell. This DNA can have a variety of different characteristics: it may be easily traceable, or it may be very susceptible to a specific chemotherapy drug, for instance.
This sort of treatment is years away. It's made more complicated by the fact that the process of getting competent cells to take up trojan DNA is time consuming, a real problem when you're working with a real person.
Web Resources On Competent Cells
The Biology Place: Competent Cells
Book Resources On Competent Cells
Molecular Biology of the Cell by Bruce Alberts et al
Molecular Biology of the Cell: A Problems Approach by John Wilson