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So it's been a while that I have been wondering how each level of speech is
called in Japanese. All I have seen, is in textbook the mention of casual,
polite, humble, or honorific speech, but I realized that I have no idea how to
say each in Japanese (not going into slang, dialects or anything). Also, how
can I recognize someone telling me it's okay to be more casual with them
(example of natural sentences I might hear)? Would it be something like
[casual speech]でいいのよ or [casual speech]使っていいよor something more along the lines
of "you don't need to be that polite"? (however you might say that).

Thank you very much.
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2016-05-31T00:50:56.830
34509
2016-06-01T13:17:19.170
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question
politeness
polite and casual language-- being asked to switch?
First of all, linguistically you are not wrong even if you used a less polite
form. However, you are impolite in-terms of culture understandings.  
(@user11589 has already gave you a very good link for reference, so I'm just
going to go simple.)

A simple way to decide which form to use is by the situation and person you
are talking to (similar to dress-code, time/place/occasion a.k.a. TPO).

  * to strangers (even if they are kids better to use) = polite 
  * to shopkeeper/staffs = polite
  * to customer = honorific
  * to people older than you = polite (or honorific)
  * to people who is higher than you in position(work) = honorific (you can transit into polite once you have build a closer relationship)

As you can see by now, most of the time using polite form will be fine.

  * friends (regardless of age) = casual
  * colleagues (position same or lower than you) = casual/polite
  * siblings = casual
  * Family members (older) = polite 
  * Family members (younger) = casual

> Also, how can I recognize someone telling me it's okay to be more casual
> with them (example of natural sentences I might hear)? Would it be something
> like [casual speech]でいいのよ or [casual speech]使っていいよor something more along
> the lines of "you don't need to be that polite"? (however you might say
> that).

As to your question, YES some people do say it. Example： タメでいいよ OR
けいごじゃなくてタメでいいよ。 (you don't need to use honorific/polite, casual is fine)

Note:  
another way to know if you can switch to casual, if you are talking to people
in similar age group and they changed into casual (from polite), you can
change into casual as well.
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2016-05-31T05:20:35.607
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2016-06-01T00:22:28.897
2016-06-01T00:22:28.897
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answer
Maki's answer covers most of your questions, so I'm only going to go into
common names for some of the different speech modes. However, as other
comments/answers have pointed out, these speech mode names don't come up in
day-to-day conversation in the way you're asking about (at least in my
experience). They're more useful in linguistic contexts (eg "how would I say
this sentence using honorific speech?")

通常口語 つうじょうこうご "casual speech" (actually, I'm not sure about this one.. haven't
run into a phrase for "casual speech" in day-to-day conversation so this is an
educated guess)

丁寧語 ていねいご "polite speech" is an umbrella term for non-casual speech (eg using
です/ます sentence endings, etc)

敬語 けいご "respectful speech" ~ in my experience, there's some variance in how
this is interpreted. Some speakers take this to mean the more formal polite
speech used in business or customer support, but some speakers take it to
include basic "polite speech" as well.

The next two are more technical, and more precise in their meanings. They
split formal "super-polite" Japanese into two major types:

尊敬語 そんけいご "honorific speech" ~ speech that "elevate" your partner/out-group
showing that you respect their greater status. eg ご覧になる instead of 見る

謙譲語 けんじょうご "humble speech" ~ speech that lowers yourself/in-group relative to
the speaking partner/out-group, showing their greater status indirectly. eg
拝見（する) instead of 見る

also, not directly related to your question but sometimes useful:

標準語 ひょうじゅんご "standard speech" eg "standard Japanese" as in "textbook Tokyo-
dialect Japanese"
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2016-06-01T13:17:19.170
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answer