Learn German: Ich esse Brot mit Butter. - I eat bread with butter.

Just like in English, when you talk about uncountable food items in a general way, you can drop the article. You are talking about bread and butter in general, not a specific loaf or stick of butter.

I know mit is a dative preposition. How does that affect this sentence?

The word mit always requires the dative case. Because we dropped the article for Butter, you don't see the case change here. But if you wanted to say I eat bread with the butter, it would become mit der Butter since Butter is a feminine noun.

Is the verb essen irregular?

Yes, but it looks regular in this sentence. The ich (I) form is simply ich esse. However, it changes its stem vowel for other people: you would say du isst (you eat) and er/sie/es isst (he/she/it eats).