“Pleuvoir” is a 3rd group verb and is intransitive (intransitif in french).
It will be use (only with the ‘avoir’ verbs) at the third persons for the most of the time (Indicatif,Conditionnel & Subjonctif).

You can compare voirverb with pleuvoir verb to see that there is only a third person for pleuvoir conjugation.

You may use also an impersonal version:
C'est pluvieux aujourd'hui. (which still use a the 3rd person)

This kind of verb is called an impersonal verb (verbe impersonnel). Impersonal verbs have no subject (no “person” doing the action). They exist in many Indo-European languages (and possibly others). In some languages, such as Latin and Spanish, this is easy: you can omit the subject. Other languages, such as French and English, require a subject, so impersonal verbs have a dummy subject which is a third person pronoun that doesn’t actually stands for anything.

[Latin] Pluit. [Italian] Piove. [Spanish] Ilueve.
[French] Il pleut. [English] It is raining. [German] Es regnet.

The pronoun il is the only one in (mainstream) French that can be impersonal. Ce and ça always have a referent. (There is a fairly rare variant “ça pleut”. Many French people consider it incorrect.) I don’t know why; it may be one of those many language questions where the answer to “why” is that it just happened like that historically.

Many verbs can be impersonal some of the time. Some weather verbs such as pleuvoir (rain), neiger (snow), grêler (hail), etc. are pretty much exclusively impersonal. The only verb I can think of that is exclusively impersonal and not related to weather is falloir: il faut + infinitif expresses obligation (either physical or moral obligation). There are many more verbs that can be constructed either personally or impersonally.

Impersonal verbs have tenses and moods like any other verb: il neigeait, il neigera, il neigerait, qu’il neige, il a neigé, en neigeant, etc.