I understand that the human brain consists of many cells working together but could a single celled life form evolve something like a brain with similar intelligence to the human brain? I was thinking of a situation in which a planet has a highly advanced civilization of single celled organisms.
I don't think there's enough to a single Earth-model cell to encode the amount of information that intelligence rests on. Although there are the oddities we know as slime moulds. One might call them single cell organisms but that would be rather perverse. They're rather a form of life that has lots of cell nuclei floating around in a quantity of protoplasm bounded only by an external membrane. So elsewhere, life might not involve cells in the same way it does here.
Another model is a colony of eusocial creatures like bees or termites. The creatures are not particularly smart and may not be capable of life separated from the colony. The colony should be viewed as one individual far smarter than its ( non sentient?) members. SF of note: Orson Scott Card Ender's Game and Charles Stross missile gap.
Swarms of honey bees can make good decisions despite the fact that individual bees are rather stupid (Honeybee Democracy, Thomas D. Seeley). Single celled organisms can, in theory, develop a similar type of collective intelligence without actually merging into a multicellular organism. This idea has been used in this book.
I once had some thoughts about a society of intelligent unicellular animals (I never got far enough to decide if they were more like bacteria or amoebae). The idea was that a great number could link themselves together and act as neurons, enabling some sort of thinking. They would then attract more unicellulars to act as muscles, and build (in my case a city which baffled the explorers who found it -- sophisticated and well-maintained but apparently empty). When they were finished, they went back to being unicellulars.
The major problems I encountered were (1) why/how they decided to combine -- in other words, which unicellular decided to start a brain; (2) how they could have evolved (once you've mastered the trick of multicellular life, why give it up?); and (3) how they could transmit knowledge to their successors.
But if it helps you in building your world, you're welcome.