We consider a broadband over-the-air computation empowered model aggregation approach for wireless federated learning (FL) systems and propose to leverage an intelligent reflecting surface (IRS) to combat wireless fading and noise. We first investigate the conventional node-selection based framework, where a few edge nodes are dropped in model aggregation to control the aggregation error. We analyze the performance of this node-selection based framework and derive an upper bound on its performance loss, which is shown to be related to the selected edge nodes. Then, we seek to minimize the mean-squared error (MSE) between the desired global gradient parameters and the actually received ones by optimizing the selected edge nodes, their transmit equalization coefficients, the IRS phase shifts, and the receive factors of the cloud server. By resorting to the matrix lifting technique and difference-of-convex programming, we successfully transform the formulated optimization problem into a convex one and solve it using off-the-shelf solvers. To improve learning performance, we further propose a weight-selection based FL framework. In such a framework, we assign each edge node a proper weight coefficient in model aggregation instead of discarding any of them to reduce the aggregation error, i.e., amplitude alignment of the received local gradient parameters from different edge nodes is not required. We also analyze the performance of this weight-selection based framework and derive an upper bound on its performance loss, followed by minimizing the MSE via optimizing the weight coefficients of the edge nodes, their transmit equalization coefficients, the IRS phase shifts, and the receive factors of the cloud server. Furthermore, we use the MNIST dataset for simulations to evaluate the performance of both node-selection and weight-selection based FL frameworks.