This is a very broad question. There are many kinds of single-celled, or "unicellular" organisms, from all three domains of life (Archaea, Bacteria, Eukarya). How a unicellular organism functions in general depends on what kind of organism it is. However, one thing that connects all of them is how they reproduce, which is an essential part of the definition for life. Most unicellular organisms reproduce asexually, through a variety of methods, such as budding, fission, or fragmentation.

In terms of other kinds of functioning, factors like whether the cell is prokaryotic vs. eukaryotic will come into play. This classification tells us about the structure of the cell. Prokaryotic cells lack membrane-bound organelles, including the nucleus, mitochondria, and chloroplasts. This has implications for how cellular processes, such as DNA replication, are carried out. Cells can further be classified as autotrophic or heterotrophic -- this classification tells us about how organisms obtain their energy. Autotrophic cells do not rely on other organisms to obtain their energy. They undergo processes such as photosynthesis or chemosynthesis to sustain themselves. On the other hand, heterotrophs rely on other organisms to get their energy.

So to summarize, there are many ways in which unicellular organisms can function. It depends on what kind of organism they are, but they all have to do the same things in order to survive: they need to reproduce and obtain energy somehow. I realize this is vague, but I hope this helps a little!