Which of the following statements is correct ?

To determine the correct statement among the given options, let's analyze each option step-by-step based on biological definitions and concepts:

1. Analysis of Option 1: "Fusion of two cells is called Karyogamy."
In sexual reproduction of fungi and other organisms, the fusion of two cells (specifically their protoplasts) is called plasmogamy. The term karyogamy specifically refers to the fusion of the two nuclei, which happens after plasmogamy. Therefore, this statement is incorrect.

2. Analysis of Option 2: "Fusion of protoplasms between two motile on non-motile gametes is called plasmogamy."
Plasmogamy is defined as the fusion of protoplasms (the living contents of a cell, including cytoplasm and organelles but excluding the cell wall) between two gametes, which can be either motile or non-motile. This is the initial stage of syngamy/fertilization. Therefore, this statement is correct.

3. Analysis of Option 3: "Organisms that depend on living plants are called saprophytes."
Organisms that depend on living plants (or animals) for their nutrition are called parasites. Organisms that depend on dead and decaying organic matter for their nutrition are called saprophytes. Therefore, this statement is incorrect.

4. Analysis of Option 4: "Some of the organisms can fix atmospheric nitrogen in specialized cells called sheath cells."
Certain cyanobacteria (like Nostoc and Anabaena) can fix atmospheric nitrogen in specialized, thick-walled cells called heterocysts, not sheath cells (sheath cells generally refer to protective or structural outer layers). Therefore, this statement is incorrect.

Conclusion:
Comparing all the statements, only the second statement regarding the definition of plasmogamy is biologically accurate.

Hence, the correct option is: Fusion of protoplasms between two motile on non-motile gametes is called plasmogamy.