Multinomial Naive Bayes models each class as a distribution over words. To classify, it sums log-probabilities of the document's words under each class plus the class prior, and picks the largest.
Laplace smoothing (add-one) prevents a single unseen word from zeroing out a class probability. Work in log space to avoid numeric underflow.
It trains in one pass, works with little data, and is a classic spam filter. Logistic regression on TF-IDF usually edges it out on accuracy.
Going deeper
Complement Naive Bayes (ComplementNB) is more robust on imbalanced text classes. NB-SVM (Naive Bayes features into a linear SVM) was a famously strong sentiment baseline.
Naive Bayes trains incrementally (partial_fit), useful for streaming classification such as spam filtering.