Standardization refers to scaling a variable that has a Gaussian distribution such that it has a mean of zero and a standard deviation of one.
Normalization refers to scaling a variable that has any distribution so that all values are between zero and one.
It is possible to normalize after standardizing a variable.
Generally, normalization and standardization are data scaling methods, and there are other methods that you may want to use.
Scaling methods are often appropriate with machine learning models that learn or make prediction on data using the distance between observations (e.g. k-nearest neighbors and support vector machines) and methods that calculate weighted sums of inputs (e.g. linear regression, logistic regression, and neural networks).
If you are still in doubt as to whether you should standardize, normalize, both, or something else, then I would recommend establishing a baseline model performance on your raw data, then experiment with each scaling method and compare the resulting skill of the model.
For more information on how to standardize data and normalize data, see the tutorial:
How to Use StandardScaler and MinMaxScaler Transforms in Python