Pure carbon planets don’t exist, but carbon planets might and are probably rare in the universe, but could have truly dramatic scenery with immensely tall mountains.
We don’t know a carbon planet yet. They are hypothetical and might form in systems with a high carbon-to-oxygen ratio. Therefore, to look for them, it would be smart to examine stars with such an elemental composition. Our planets formed from an oxygen-rich protoplanetary disk, and Earth is composed of 30 to 47% of this element, while carbon is only 0,025 to 0,03%. If we ever find a carbon planet, it is expected to have a carbon-to-oxygen ratio close to 1.
This would come with dramatic characteristics. Although a wide range of minerals could be found on such worlds, there could be diamond mountains and graphite sands. Carbon in this form is very tough. On Earth, the highest mountains are just under 9 km tall, but on a similarly sized carbon planet with oceans and continents as Earth, diamond mountains could reach 20 to 25 km tall.
They wouldn’t shine with jewelry-grade sparkling diamonds. The atmosphere would be expected to be filled with carbon compounds and hazes. With so little oxygen, there would be no water, which is composed of this element and hydrogen. Instead, depending on temperature and other conditions, there could be oceans of hydrocarbons such as methane, ethane, or heavier oily ones.
Rains of these liquids would cover all the landscape, and few places would be expected to have shiny and exposed diamond mountains, if at all such places would exist. Maybe around waterfalls and flowing water, there could be some.
We are not sure if plate tectonics is possible on carbon planets. A diamond crust would be more rigid, and we are not sure if oceans made of molecules other than water could work for this process. Also, diamonds conduct heat well, and the crust on carbon planets might become too thick and rigid due to faster internal cooling. Maybe super-Earth-sized carbon planets would be more likely to host plate tectonics.
This means that complex life might be less likely. Simple microbial cells could exist in a different biochemistry with a different non-polar solvent. Their metabolism might be slower, and they could rather form greasy biofilms. If such worlds could nevertheless host animal and plant-like organisms, and a technological civilization could emerge, interstellar travel might be easier for them. They would live longer, and because of their slow metabolism, they would perceive the passage of time more slowly than we do. Hundreds or thousands of years' journeys to nearby stars might be less difficult for them.
Less than 10% of stars in our galaxy are carbon-rich. Although such planets might be rare, there should be some in our neighborhood, and hopefully, one day we will find them and visit them, at least via our robotic missions that could send footage back to Earth. It would be immensely fascinating.