In a nutshell: fitting is equal to training. Then, after it is trained, the model can be used to make predictions, usually with a .predict() method call.
To elaborate: Fitting your model to (i.e. using the .fit() method on) the training data is essentially the training part of the modeling process. It finds the coefficients for the equation specified via the algorithm being used (take for example umutto's linear regression example, above).
Then, for a classifier, you can classify incoming data points (from a test set, or otherwise) using the predict method. Or, in the case of regression, your model will interpolate/extrapolate when predict is used on incoming data points.
It also should be noted that sometimes the "fit" nomenclature is used for non-machine-learning methods, such as scalers and other preprocessing steps. In this case, you are merely "applying" the specified function to your data, as in the case with a min-max scaler, TF-IDF, or other transformation.
Note: here are a couple of references...
- fit method in python sklearn
- http://scikit-learn.org/stable/tutorial/basic/tutorial.html
