The plain form of a verb is an uninflected or less-inflected form that is used in several roles. The term 'plain form' is used when no specific traditional term (such as present tense or infinitive) is suitable. The two languages most commonly described as having a plain form are English and Japanese. In both of these the plain form is the form that occurs in a dictionary, and other forms of the verb can generally be regarded as built up from it.

Japanese plain form

To dispose of Japanese briefly: the form ending in -u or -ru is used as (1) the main clause form of the verb in familiar (non-polite) style, and (2) the form used in subordinate clauses even in polite style. See the node Japanese verbs for details.

English plain form

uses

In English the plain form is the bare form of the verb (i.e. without any inflection at all). It is used (1) after some auxiliary verbs, (2) in clauses marked with the infinitive particle to, (3) in simple commands and suggestions, and (4) in some other clauses that are not statements of fact but unreal or desired states. Examples of these four uses:

(1) She can leave now.
(2) I want her to leave now.
(3) Leave now.
(4) We insisted that she leave now.

By contrast, inflected forms are used in simple main clause statements, in other kinds of subordinate clause, after prepositions, and so on:

(5) She leaves tomorrow.
(6) She left yesterday.
(7) I will be unhappy if she leaves.
(8) I admire her for leaving.

form v. function

In more heavily inflected languages related to English, such as German or Latin, there is not a single form that covers all of situations (1) to (4), and this was case a thousand years ago in Old English too. In such languages there is an infinitive form for use (1) and perhaps (2), an imperative for use (3), and a subjunctive for use (4) and perhaps (2). But Modern English lacks these forms: it has a plain form instead (historically derived by fusion of the different forms). We can still talk about distinct 'clause types', saying that her to leave now is an infinitive clause, Leave now is an imperative clause, and that she leave now is a subjunctive clause. But these are functions of the verb, or forms of the clause, not forms of the verb.

the present tense paradigm

A problem arises when we turn the present tense inflected form leaves into plural, or into first or second person:

(5) They leave tomorrow.
(7) I will be unhappy if you leave.

The plain form and present tense form are only distinct in the third person singular. Do we then say the plain form is also sometimes used in clauses of the type (5) to (8), depending on person and number? We certainly could. If we are strict about forms, they are the same form. One argument against this is that one verb, be, uniquely has different forms here (they are v. that they be); however, be also has distinct first person forms (I am, I was), which no other verb has, and a distinct subjunctive (if she were), which again no other verb has. So be needs special treatment anyway, and shouldn't be a model for other verbs.

The authors of the CGEL (see reference below) choose to treat the present tense non-3sg. as grammatically distinct from the plain form, though phonetically identical. The present tense is a well-defined function, and it makes sense to have all its forms belong to the same paradigm, rather than having one belonging to it and the other not especially part of it. But this is arguable, and possibly unimportant. One theoretical argument for maintaining the distinction is the status of agreement: present tense leave agrees with its subject in person and number (to some extent), whereas plain form leave lacks agreement entirely (as do the two other non-finite verb forms, the gerund/present-participle leaving and the past participle in she had left).

frozen subjunctives

A small group of exceptional constructions have a plain form at the head of a main clause. These are generally either optative in meaning (expressing a wish) or concessive, and include "So be it", "God save ...", "God bless ...", "Be that as it may", "Come what may", "Be it ever so humble", and so on. Historically these are subjunctives. The pattern is no longer productive: it is frozen. These particular expressions have to be learnt like words, and the grammar does not allow new, similar constructions to be freely made. Even substituting Allah, Jesus, or Jupiter for God sounds forced: more like a pun than regular grammar. So, grammatically, this does not really count (any more) as a situation where plain forms are allowed.

Reference. The definitive grammar of English is now Huddleston and Pullum, 2002, The Cambridge Grammar of the English Language. In a couple of minor points I have used very slightly different terminology to theirs.