Answer
You can probably already guess the answer to your question: Unicellular organisms can reproduce, of course, otherwise they would never have been here with so many. The reproduction of a single-celled animal is (usually) asexual, and works in a slightly different way than, for example, reproduction in humans.
Most unicellular organisms do not reproduce by having sexual intercourse, but by growing sufficiently and splitting into 2 single-celled organisms. In eukaryotes (cells with a cell nucleus) this process is called “mitosis”. In this process, the genes are duplicated and evenly distributed over the cell so that each new cell receives the correct number of genes when it is split.
Mitosis is not only a very important process for single-celled organisms, it is also very important for us, multicellular organisms. All cells in our body are created by the duplication, or mitosis, of a single fertilized egg. When making eggs and sperm cells, our body uses “meiosis”, a process similar to mitosis, but in which a cell divides into 4 daughter cells after deduplication, in which each daughter cell receives exactly half of the genes of the original cells. If then such a “haploid” (cell with half of chromosomes, normally each of our cells has 2 versions of each chromosome and we call them “diploid”) egg cell and haploid sperm cell are brought together in a fertilization, the process of start mitosis from the beginning and grow a complete multicellular organism. During these processes of meiosis and fertilization, the genes of the parents are scrambled, so that the child receives a good mix of the genes of its parents. This is different in mitosis, where a child gets exactly the same mix of genes as its parent.
There are of course a few exceptions to every rule in biology. For example, there are green algae (Chlamydomonas), which are (haploid) unicellular organisms with a “sex” (we do not speak of males or females here, but of + and -). The cells can reproduce through mitosis, but if the environmental conditions are right, a “male” algae and a “female” algae can fuse (similar to fertilization), and undergo meiosis to make 4 daughter cells together, with sexual reproduction. These algae are very important in biological research because they can tell us how sexual reproduction originated on Earth. We also learn from these algae how important sexual reproduction is in evolution. By mixing the parental genes during sexual reproduction, many more differences between individuals can arise, and the species is thus better able to adapt to changes in its environment.