I would like to know what is term for a verb when is with "to" (for example "We need "to do" it.) and a verb that is without "to" (for example: "We do it everyday"). Now I read the article infinitive on wikipedia but it is not clear to me how differentiate between them.
I do the work every day. do = is the verb, the main verb in the sentence, aka an action verb.
I need to do the work today. a to-infinitive with the verb to need [something].
Verbs with to-infinitives We use the to-infinitive after certain verbs (verbs followed by to-infinitive), particularly verbs of thinking and feeling: [there is an entire list here.]
Need, like want, love, hate, is considered a verb of feeling: I need to leave now.
[The examples would be the same in American English.]
They are often referred to as "infinitive with 'to'" and "infinitive" (or "bare infinitive").
I prefer to call "do" the "base form" (infinitive is one of its uses).
Edit: Correction: the form in your second example is not the "bare infinitive"/"base form", but the (not third-person-singular) present form. For nearly all verbs this is the same as the base form.
Verbs that aren't infinitives are finite.