need to understand infinitive

Question

What is the easiest way to understand what an infinitive is?
How do I know which verb in which sentence is an infinitive?
For example, let us take this website:
Infinitive
This is the example I am interested in:

1. I need to run every day. (The infinitive form with the word to is called the full infinitive or to-infinitive.)
2. I must run every day. (After certain verbs, the to is dropped (more on this below).)
3. I run every day. (This is not in the infinitive form. This is a finite verb, i.e., a verb functioning as the main verb.)

How do I differentiate between example 2 and example 3?
How do I know 3 is not an infinitive but 2 is ?
If 2 had been I must run to school every day would that make it a non-infinitive?

This is probably a completely different question, but I will put it here anyway.
When comparing with other languages, if it says infinitive, should I consider it to have the same meaning as the English infinitive, or can it have a different meaning, according to the language?

Let us take the example of Korean. This Wikipedia page has an entry for infinitives in Korean. If I understood correctly, "infinitive" is the form of the verb that is unchanged according to tense, voice or anything similar. (So watch may be an infinitive but watched is not). In Korean, the unchanged form of a verb is called root (example, watch = 보다). This root changes form (prefixes, suffixes etc, based on negation, formality, tense, etc). Can this root be called an infinitive as well? (in my opinion, probably not, as the explanations in the wikipedia page I linked seems to use root and infinitive as different entities, but how do I explain the difference)?

Answer 1

To answer the last question first, Infinitive is a term from Latin grammar.
It refers in Latin to one of several tenseless verb forms.

Etymologically, infinitive means 'unending' (just like infinite); here the reference is to lack of tense.
In other languages, infinitive is often used to describe one, or more, untensed verb forms.
The Latin terms Participle, Gerund, and Gerundive are also in use for untensed verb forms.
What these terms mean, in each language, has to be determined individually. There is no consistency.

The rest of the question is about English infinitives, and here the trick is knowing
that there are two different meanings for infinitive in English grammar.
If you keep them straight, you won't be easily confused.

1. Infinitive can refer to the infinitive form of the verb, e.g, be, run, go, see, feel, have, admit, ask
   Every verb in English (except defective verbs like born and modal auxiliaries) has

• Three Principal Parts: be, was, been; run, ran, run; go, went, gone; see, saw, seen;
  feel, felt, felt; have, had, had; admit, admitted, admitted; ask, asked, asked.
  These three verb forms are called, in order, the Infinitive, the Past, and the Past Participle.

2. Infinitive can also refer to infinitive clauses of various kinds, which are subordinate clauses
   that use infinitive verb forms, instead of present, past, or participial forms.

• For him to err is human ~ He wants to leave early ~ She's the one to see about that.
  These are normally marked with the for...to complementizer, though the for that
  marks the subject of the clause is frequently absent, along with the subject itself.

As @YellowSky has pointed out, most infinitive clauses are governed by the matrix predicate.
There are a number of auxiliaries that must be followed in construction by an infinitive verb form.

All the modal auxiliary verbs (including the NPI semimodals need and dare,
as well as the periphrastic modals like hafta, wanna, oughta, able to, and need to)
must be followed immediately by an infinitive verb form, and the same is true of do
in its Do-Support and Emphatic senses.

• Do you like ice cream? but not *Do you liking/liked ice cream?
• I did take it out! but not *I did taking/took it out.

These infinitive verb forms are part of a verb phrase and don't use the complementizer to, because they don't introduce complements, just the next verb in a verb chain. See the Verb Phrase Guide for details.

Executive Summary:

Infinitive clauses are very complex, and they use infinitive verb forms.
Infinitive verb forms are used in infinitive clauses (usually marked by to);
infinitive verb forms also occur with auxiliary verbs in verb phrase constructions
(usually without to).

Answer 2

Infinitives are rather different in different languages, and some languages, like Bulgarian, don't have infinitives at all. In some languages infinitives have a special suffix and are easily recognized, e. g. in Russian the most typical infinitival suffix is -ть, like in писать 'to write', читать 'to read', etc.

In English the infinitives are in most cases marked by the infinitival particle 'to' (to-infinitive), but sometimes they are not (bare infinitive), still the list of cases when they are not mark is pretty short. The English unmarked infinitives follow another verb. Here is the list of the verbs that are always followed by an unmarked infinitive, infinitives are marked in bold:

1. The nine defective verbs: can, could, shall, should, will, would, may, might, must: I cannot do that.
2. The verb 'let': Let me help you.
3. The verb 'help': He helps me do my homework. (Nowadays many use the to-infinitive after 'help'.)
4. The verb 'make' in the meaning 'to cause': He made me cry.
5. The verb 'do' (Past 'did') used to form questions, negative sentences, and emphatic sentences in the Present Simple Tense and in the Past Simple Tense: Do you know him? (question), I don't know him. (negative), I do know him! (emphatic);
6. After 'needn't', 'need not' and questions with 'need [subject]': You needn't go there.

Unlike many languages, the English infinitive has different forms, 4 for an intransitive verb (they don't have Passive forms), and 8 for a transitive one. English infinitives can change for voice (Active: to write, Passive: to be written), for aspect (Non-Continuous: to write, Continuous: to be writing), for perfectness (Non-Perfect: to write, Perfect: to have written), and there are Perfect-Continuous infinitives: to have been writing. So, you got it wrong, the infinitives can change after the grammar categories.