Her mother died at 85.
It's called the "to-infinitive" because the first word is "to". For example, "Her mother lived to be eighty-five."
Why it's called "infinitive"
The word "infinitive" is taken from Latin. The Romans, when describing Latin grammar, called some of their verb forms modus infinitivus, which means "the indefinite way", because unlike other verb forms, the infinitive ones did not indicate person and number. The infinitive is like an abstract form of the verb. In "John would like to buy some doughnuts", the verb "buy" doesn't agree with John. It names the action of buying without asserting or denying that someone is buying something or specifying a time when the buying happens. That's why the Romans felt that it was "indefinite".
In the sentence "John buys doughnuts during the week", "buy" is called a "finite verb" because it agrees in number with John, it specifies a time when the buying happens, and most importantly of all, it asserts that John buys the doughnuts at that time. The word "finite" comes from Latin finitus, which means limited or bound.* A finite verb is bound—that is, tied to—a subject, about which the verb makes a claim. Finitus comes from the same Latin verb as the English word "finish". You can think of an infinitive verb as "unfinished" in a certain way.
How to spot an infinitive
English verbs work very differently from Latin verbs, though. Latin verbs have about 150 different forms, which simultaneously indicate person, number, tense, mood, and voice. English verbs have only a few forms, like buy, buys, bought. The "bare" form of the verb serves many purposes: it's the infinitive, the imperative, the first-person singular and plural, the second-person singular and plural, and the third-person plural. So, English uses subtle tricks to make clear when a verb is meant as an infinitive.
For example, in:
Make John buy me a doughnut.
"buy" is an infinitive. How do you know? You just have to know that the verb make can take a direct object which is the subject of an infinitive verb that follows it. (The infinitive in that sentence has a subject, John, but it doesn't agree in number or person with that subject.) Let and help follow the same pattern.
Modal verbs put the following verb into the infinitive, like this:
John would buy doughnuts if he had enough money.
Don't buy yesterday's doughnuts.
More often, English sticks the word "to" in front of infinitives, which helps make clear that the verb is not being used in a finite way. For example, in:
It's good to be the king.**
You need the "to" to indicate that "be" is to be understood in the abstract. The sentence means the same as "To be the king is good."
Now here is where it starts to get hard—very hard. English also uses the gerund to mean the action of a verb in the abstract, without needing to agree with a subject's person and number. Often you need a verb in such a "non-finite" form to make a combination with another verb, like this:
John meant to buy doughnuts but he forgot to buy them.
John considered buying doughnuts and decided not to buy any.
John imagined buying doughnuts, he dreamed of buying doughnuts, and he wished to buy doughnuts.
In these combined verbs, the first verb is finite and the second verb is either a gerund or a to-infinitive. How do you know whether to use a gerund or a to-infinitive? It depends entirely on the first verb. Some verbs take a gerund and some take a to-infinitive, and there is no rule. As far as I can tell, there is no pattern whatsoever. If you want to feel discouraged, look at this list. You just have to learn one verb at a time, to find out whether it requires a gerund or a to-infinitive when combined with a second verb.
I haven't even told you all of the complexity. I don't know how anyone can possibly learn when to use a to-infinitive, when to use a bare infinitive, and when to use a gerund. Somehow, though, people do. Somehow, native speakers always know which is the correct form. The only thing I can recommend is to have a lot of patience with yourself as you learn. Just imitate what you read and hear, and eventually, somehow, you will find yourself choosing the correct verb form. (Why isn't it "find yourself choose", by analogy with "make yourself choose"? There is no reason.)
*The word "infinite" comes from the "limited" sense of finitus: the Latin roots mean "without limit" or "without end".
**Source: The movie History of the World, Part I.