The base variety of verbs are the kind you see in a dictionary, like flirt, cuddle, argue, break up. Slap to in front of them, as in to flirt, to cuddle, to argue, to break up, and you get your infinitive.
You "split" an infinitive by placing a word between the to and the actual verb.
Traditionally, you weren't supposed to split infinitives. Why? Because of some archaic Latin rule. But, if you haven't noticed, we've transitioned from Latin to English, and English isn't Latin.
It's now okay to split infinitives, especially if you'd have to perform grammatical gymnastics to avoid the split. If anyone disagrees, send them our way, and we'll tell them what's up.
"Maximillian tried to legally change his first name to Maximillianus at the courthouse."
Maximillianus? We think we'd just have gone with Max, but it's none of our business. Our only business here is with the split infinitive, or an infinitive form of a verb that contains a modifying word between to and the rest of the phrase. That happens here with legally wedged between to and change.