Alzheimer's disease is largely the underlying cause of dementia due to its progressive neurodegenerative nature among the elderly. The disease can be divided into five stages: Subjective Memory Concern (SMC), Mild Cognitive Impairment (MCI), Early MCI (EMCI), Late MCI (LMCI), and Alzheimer's Disease (AD). Alzheimer's disease is conventionally diagnosed using an MRI scan of the brain. In this research, we propose a fine-tuned convolutional neural network (CNN) classifier called AlzheimerNet, which can identify all five stages of Alzheimer's disease and the Normal Control (NC) class. The ADNI database's MRI scan dataset is obtained for use in training and testing the proposed model. To prepare the raw data for analysis, we applied the CLAHE image enhancement method. Data augmentation was used to remedy the unbalanced nature of the dataset and the resultant dataset consisted of 60000 image data on the 6 classes. Initially, five existing models including VGG16, MobileNetV2, AlexNet, ResNet50 and InceptionV3 were trained and tested to achieve test accuracies of 78.84%, 86.85%, 78.87%, 80.98% and 96.31% respectively. Since InceptionV3 provides the highest accuracy, this model is later modified to design the AlzheimerNet using RMSprop optimizer and learning rate 0.00001 to achieve the highest test accuracy of 98.67%. The five pre-trained models and the proposed fine-tuned model were compared in terms of various performance matrices to demonstrate whether the AlzheimerNet model is in fact performing better in classifying and detecting the six classes. An ablation study shows the hyperparameters used in the experiment. The suggested model outperforms the traditional methods for classifying Alzheimer's disease stages from brain MRI, as measured by a two-tailed Wilcoxon signed-rank test, with a significance of <0.05.