Can anybody let me know the definition of these terms? I know we solve this for Beta but I want to have the definition
The function you are trying to minimize when using ridge regression consists of two parts, a standard loss function that describes how well the model fits the data (in your example this is the mean-squared error) and a penalty term. The penalty term is simply the summed absolute values of the model's parameters multiplied by lambda, which is a hyperparameter of the ridge regression. This describes how much of an impact the penalty term has on the total loss, a large lambda means that the model will likely favor a simpler model with smaller (in value) parameters. The model therefore has to make a trade-off between the two, improving the fit with regards to the data while making sure to keep the values of the parameters relatively small to prevent the penalty term from becoming too big.