The short answer is yes, there are single-celled organisms that can reproduce without another "partner". Probably the most famous example is that of bacteria.
What you're talking about is known as asexual reproduction. In bacteria, the process is known as binary fission, where one bacterium (known as the parent cell) divides into two organisms (known as daughter cells). These two cells will be genetically identical (barring any mutations) to the parent cell, because there has been no combination (for lack of a better term) of DNA, as there is in sexual reproduction.
The process works the same way it does in cells in multicellular organisms, such as humans: DNA "unwinds" (again due to lack of a better term) in the center of the cell, complementary bases are added to each of the now-separate strands (adenine with thymine, guanine with cytosine), the other parts of the cell are duplicated, and, gradually, the bacterium splits in two. The process can be very quick, compared to the nine months humans spend in the womb.
Asexual reproduction is not unique to single-celled organisms, though. While it is prevalent (and easy) in bacteria, some multicellular organisms reproduce asexually. Some plants are capable or this via self-pollination, and there have been some reports (albeit rather dubious) of certain "common" animals reproducing asexually.
There are advantages and disadvantages to asexual reproduction. One advantage is obvious: no mate necessary. As long as an organism has enough energy and nutrients to reproduce, it can. Also, the process is quick, and easy. There are some disadvantages, though. The big one is that there is no recombination from multiple DNA sources - that is, the resulting organism will have exactly (again, barring mutations) the same genes as its parent - for better or for worse. If an asexually-reproducing organism has a defect that still allows it to reproduce, chances are, its offspring will have that defect.