The infinitive is the unconjugated, or root, form of the verb. For example, to be in English, être in French, esse in Latin; or to love in English, aimer in French, amare in Latin. Notice how the infinitive in English is two words, but only one word in French or Latin.
When Latin was a high-status language, grammarians used to try to force the rules of Latin onto English, a language with a rather different grammar. One such rule they invented was "do not split an infinitive" -- that is, do not put anything between the to and the rest of the verb. Their reasoning seems to be that, since it is impossible to split the single-word infinitive in Latin, one shouldn't split it in English, either. So, something like to boldly go is beyond the pale, according to these Latin grammarians.
"It is exceedingly difficult to find any authority who condemns the split infinitive - Theodore Bernstein, H. W. Fowler, Ernest Gowers, Eric Partridge, Rudolph Flesch, Wilson Follett, Roy H. Copperud, and others too tedious to enumerate here all agree that there is no logical reason not to split an infinitive." -- Bill Bryson, Mother Tongue, 1990