Traditional von Neumann computing architectures face fundamental limitations in energy efficiency and parallel processing, particularly for artificial intelligence applications. This research investigates neuromorphic computing systems that mimic brain neural networks using spiking neural networks (SNNs) and memristive devices. We designed and fabricated a 64x64 memristor crossbar array implementing leaky integrate-and-fire neurons, achieving 1000x lower energy consumption compared to conventional processors for pattern recognition tasks. Benchmark tests on MNIST and CIFAR-10 datasets demonstrated 94.2% and 87.6% accuracy respectively, with inference power consumption of only 2.3 mW. The study provides hardware-software co-design principles for next-generation neuromorphic systems and explores applications in autonomous robotics and edge computing.