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"