The most common form of split infinitives is when the infinitive is split by an adverb. A famous example is from the introduction to the television show "Star Trek," which announces that its quest is "to boldly go" where no man has gone before. Other examples include "I thought I was going to never get there" and "Although the poll numbers are down, I hope to successfully run for office."
Infinitives can also be split by words that are not adverbs. Most commonly, this happens with a negation. For example, you might say, "I'm hoping to not go to school today" or "I want to not fail." The other common non-adverbial word that is used to split infinitives is "all." For example, "all" splits the infinitive in the sentence "I want you to all be there tomorrow."
Avoiding splitting your infinitives is rather easy. Start by identifying the infinitive in the sentence, if there is one. If an adverb is splitting it, simply move the adverb to a different position in the sentence. In the example, "I want to quickly finish my homework," put the adverb at the end: "I want to finish my homework quickly." Sometimes it makes sense to put the adverb somewhere else. For example, in "I want to badly go to the party," rewriting it as "I want badly to go to the party" works well.
As a general rule, leave a split infinitive intact if changing it sounds awkward or changes the meaning of the sentence. There is a difference, for example, between the meanings of "I want not to go" and "I want to not go," with the latter sounding more emphatic. Similarly, "I want to really win" suggests that the speaker wants a decisive victory, whereas "I really want to win" suggests that he has a large amount of desire for a regular victory.