Integration Services uses transformations to manipulate data during an ETL dataflow. Transformations can be used in Data Flow Tasks, between data source and destination components, or other transformations.

The Row Count transformation simply counts the records flowing through a data path. It only changes one thing: the value of a numeric variable which is assigned to its VariableName property:

If the same variable is assigned to more Row Count transformations, the variable will be increased with all the record counts.

The Row Count transformation have one input and can have an output as well. The output is not mandatory.
The Row Count is a row transformation (also known as synchronous transformation), because it reuses existing buffers and does not require data to be copied to a new buffer to complete the operation.

For more detailed information on this transformation, please refer to this MSDN article.
For the complete list of SSIS transformations, check this Wiki article.