Elastic Net is a simple mix of both Lasso and Ridge.
The cost function would be like the image above. If you set the ratio r to 0, it would be the same as the Ridge Regression Cost Function. If you set it to 1, it would be the same as the Lasso Regression Cost Function.
How to decide which method to use..?
As a default, you should go for Ridge. If only few features seems to be useful, you should try out Elastic Net. Lasso tend to behave eratically when several feature are strongly correlated.
References:
Hands-On Machine Learning with Scikit-Learn, Keras, and TensorFlow, 2nd Edition
