You have already met each plain form: the dictionary form on the verb groups page, ない on negation and た on the past tense.
Short answer: every polite form has a plain twin. Verbs have four: the dictionary form (食べる), ない (食べない), た (食べた) and なかった (食べなかった). Adjectives drop です, and nouns use だ or nothing. The plain form is not rude: it is how friends and family talk, how most books and newspapers are written, and it sits inside polite sentences before words like と思います (I think) and before nouns.
The four plain forms of a verb
One verb from each group across all four forms.
Verb
Present
Negative
Past
Past negative
Group 2
食べる taberu
食べない tabenai
食べた tabeta
食べなかった tabenakatta
Group 1
書く kaku
書かない kakanai
書いた kaita
書かなかった kakanakatta
Group 1
飲む nomu
飲まない nomanai
飲んだ nonda
飲まなかった nomanakatta
Group 1
行く iku
行かない ikanai
行った itta
行かなかった ikanakatta
Group 3
する suru
しない shinai
した shita
しなかった shinakatta
Group 3
来る kuru
来ない konai
来た kita
来なかった konakatta
ある
ある aru
ない nai
あった atta
なかった nakatta
Adjectives and nouns
い-adjectives just drop です. Nouns and な-adjectives replace です with だ, but in speech だ is often dropped too, especially in questions.
い-adjective
な-adjective
noun
Where the plain form is used
Four places, and only the first depends on who you are talking to.
01With friends and familyConversation with people you are close to, and anyone younger who speaks plainly to you.明日行く？
02In writingDiaries, essays, newspapers, novels and most of the internet use plain forms.今日は雨だった。
03Inside longer sentencesBefore と思います, ので, から and many other patterns, even when the sentence ends politely.行くと思います。
04Before nounsA verb that describes a noun is always plain.昨日見た映画
Plain inside polite sentences
These sentences are polite, because they end in ます or です, but the verb in the middle is plain. This is why the plain form matters even if you always speak politely.