Correct answer: Option 'B' - int variable = 10

Explanation:
To declare a variable in the C programming language, you need to follow certain syntax rules. Option 'B' correctly follows these rules and is the correct way to declare a variable in C.

Declaration:
In C, a variable declaration consists of two parts - the data type and the variable name. The data type specifies the type of data that the variable can hold, and the variable name is used to identify the variable.

int data type:
The 'int' data type is used to declare variables that can store integer values. It is one of the basic data types in C.

Variable name:
The variable name can be any valid identifier in C, which consists of letters (both uppercase and lowercase), digits, and underscores. It should start with a letter or an underscore.

Initialization:
In the given option 'B', the variable 'variable' is initialized with the value 10. Initialization assigns an initial value to the variable at the time of declaration.

Complete Syntax:
The complete syntax to declare and initialize a variable in C is as follows:
```c
data_type variable_name = value;
```

- The 'data_type' is the type of data that the variable can hold. It can be int, float, char, etc.
- The 'variable_name' is the name of the variable, which follows the rules mentioned earlier.
- The 'value' is the initial value assigned to the variable. It should be compatible with the data type.

Example:
Here's an example that demonstrates the correct way to declare and initialize an integer variable in C:
```c
int age = 25;
```
In this example, the variable 'age' is declared as an integer and initialized with the value 25.

Incorrect options:
- Option 'A' (variable = 10) is incorrect because it lacks the data type declaration.
- Option 'C' (10 = variable) is incorrect because the assignment operator (=) should be used to assign a value to a variable, not vice versa.
- Option 'D' (variable == 10) is incorrect because '==' is the equality operator used for comparison, not for assignment.

Therefore, the correct way to declare a variable in C is option 'B' - int variable = 10.