The infinitive form of a verb is the form that is uninflected for person or tense. An inflected verb is one that is changed to show what person (first, second, or third), number of people (singular or plural) and/or time (past, present, or future) applies to the verb. For example, the bare infinitive be carries no information about who is performing the verb or whether they are performing it in the past, present or future. The inflected verb is, however, gives more specific information: it's third person, singular, present.
But if that's all a bit too complicated, you can just remember that the infinitive is the form of the verb that you would use to look the verb up in the dictionary.
In English, the infinitive is usually introduced by the particle to. We usually refer to both parts together as the infinitive. In fact, most people recognize infinitives by the fact that they're introduced by the word to:
A split infinitive is formed when a word or phrase is inserted between to and the uninflected verb, thereby splitting the infinitive.
"Our next step, Mister Palmer, is to deftly extract the bullet from this poor man's neck without disturbing the surrounding tissue."
Those who know Ducky, however, know that the professorial Brit would probably never say such a thing. Splitting infinitives is considered improper, sloppy English. It seems this is mostly because we tend to think of the uninflected verb alone as incomplete, and as the "to + infinitive" as a single unit.
So Dr. Mallard would say, instead,
"Our next step, Mister Palmer, is deftly to extract the bullet from this poor man's neck without disturbing the surrounding tissue."
"Our next step, Mister Palmer, is to extract deftly the bullet from this poor man's neck without disturbing the surrounding tissue."
or, less awkwardly,
"Our next step, Mister Palmer, is to extract the bullet from this poor man's neck— deftly, so as not to disturb the surrounding tissue."
However, there is serious debate among grammarians and linguists as to whether there really is anything wrong with splitting infinitives. As with many "incorrect" constructions, split infinitives may be used purposefully for special effect:
"The boss told you specifically to not screw up, McGee," said Tony. "So I hope you're busy not screwing up!"
One common argument against split infinitives has to do with their occurrence (or lack thereof) in Latin. It doesn't hold much water, as I've explained in comments.
Personally, I find the split infinitive to be one of the most innocuous "mistakes" a writer can make. Most of us were taught in school that it's wrong, but it's more a question of style than of correctness. A character who speaks "proper" English might be in the habit of avoiding split infinitives, and a writer should be aware of this. There are also cases in which a split infinitive just sounds weird—see my LJ cut text for an example. In most writing, however, there are more important things to worry about.