Briefly, feature is input; label is output. This applies to both classification and regression problems.
A feature is one column of the data in your input set. For instance, if you're trying to predict the type of pet someone will choose, your input features might include age, home region, family income, etc. The label is the final choice, such as dog, fish, iguana, rock, etc.
Once you've trained your model, you will give it sets of new input containing those features; it will return the predicted "label" (pet type) for that person.