From Teflpedia

A to-infinitive or full infinitive is an English infinitive that consists of two words; "to" and the base form. It contrasts with the bare infinitive. For example: "I want to play".

Note that most (all?) other Indo-European languages only have one infinitive form.

All lexical verbs have to-infinitives; but full modal verbs lack them.

To infinitives can be used as infinitives of purpose.

A to-infintive may be split to become a split infinitive.