photosynthesis - How does this diagram illustrate carbon cycling in lakes?

This shows the major biological transformations of carbon in any system (not just lakes).

On the Left Side:

• $GPP$ (Gross Primary Production) is the total amount of $C$ from atmospheric $CO_2$^† that is reduced into organic molecules during the calvin cycle of photosynthesis. This is the process performed by photosynthetic organisms like green plants and algae.

_{† atmospheric $CO_2$ diffuses into water and develops into several species of inorganic carbon molecules that plants and algae can use but in general you can think of this carbon as being very closely related to atmospheric $CO_2$}

• $OM$ is a pool of organic matter, which is a general term for reduced carbon molecules. $OM$ can be living organism tissues (e.g., a fish), dead tissues (e.g., senescent leaves) or biologically generated organic molecules (e.g., sugar that leached from a plant). $OM$ is the principle pool of energy and raw materials for all living things on Earth so this is a really important pool of stuff.

Since $GPP$ is taking $CO_2$ and converting it to $OM$ as the left side progresses, the amount of $CO_2$ in the atmosphere goes down and the amount of $OM$ in the biosphere goes up.

On the Right Side:

• $R$ (respiration) is the oxidation of reduced carbon-containing molecules for the purpose of extracting usable energy. This may be familiar as "cellular respiration" but there are other metabolic pathways that accomplish the same goal. In the end when the $C$ in the $OM$ is oxidized and the energy released the $C$ that was in the $OM$ gets converted into $CO_2$.

Note that this is the opposite of the Left Side, so as this occurs, the amount of $OM$ in the biosphere goes down and the amount of $CO_2$ in the atmosphere goes up.

If the Left and Right sides are balanced, then there is no net change in the size of the $OM$ pool or the amount of $CO_2$ in the atmosphere. However, note that over relatively short time scales these can be very out of balance. Any time plants or algae are growing (i.e., gaining mass as $OM$) the Left Side > the Right Side or $GPP$ > $R$. Once those plants die, they accumulate in the system as dead $OM$. The decomposition of accumulated dead $OM$ by microbes will generally make the Right Side > Left Side or $R$ > $GPP$.

This picture from Wikicommons shows the same cycle in a more realistic setting.