Plain forms of verbs.

[She-Lich]

All right, so just when I got done learning words like arukimasu, hashirimasu, hanashimasu, nemasu, the next chapter introduces me to the "plain" form of these verbs: Aru, hashiru, hanasu, neru.

What are "plain" verbs? What does that mean? When is it appropriate to use non-plain verbs, and when is it appropriate to use plain verbs? Are they interchangeable? Someone please explain or I'll be stuck on this chapter forever.  

[Father Harlot]

Plain form of verbs is also known as the dictionary form, so you'd find it like that in dictionaries.

Plain form is for several things, one of which is casual speech.
-masu and its conjugations are more polite, and show some distance.

It's also used for other grammar purposes like in things you need to do, suggesting someone should do something, and more.  

[She-Lich]

In the sentence structure, would it be in the same place as the more formal version?  

[Father Harlot]

Yes it would.