Question Analysis:
The question asks us to determine the number of flip-flops required to build a binary counter circuit that can count from 0 to 1023. A binary counter is a digital circuit that can count in binary from 0 to a specified maximum value. To determine the number of flip-flops required, we need to consider the binary representation of the maximum count value.

Binary Representation:
To count from 0 to 1023, we need a 10-bit binary representation since the maximum count value is 1023. The binary representation of 1023 is 1111111111.

Explanation:
A binary counter circuit can be built using D flip-flops. Each flip-flop represents a bit in the binary counter. To count from 0 to 1023, we need 10 flip-flops, each representing a bit position from the least significant bit (LSB) to the most significant bit (MSB).

Counting Process:
The counting process involves incrementing the binary value by 1 for each clock cycle. The LSB flip-flop toggles on each clock cycle, representing the least significant bit of the binary count. When the LSB flip-flop toggles from 0 to 1, it carries over to the next bit position, causing the next flip-flop to toggle. This process continues until the MSB flip-flop toggles, representing the most significant bit of the binary count.

Example:
Let's take an example to illustrate the counting process. Initially, all flip-flops are set to 0. The first clock cycle increments the count to 1, the second clock cycle increments it to 2, and so on. When the count reaches 1023, the next clock cycle resets the count to 0, completing the counting process.

Conclusion:
To build a binary counter circuit that can count from 0 to 1023, we need 10 flip-flops, each representing a bit position in the binary count. Therefore, the correct answer is option 'C'.