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It has to be prety basic but I never understood how it is conventionally explained.

Usually a pop song has a fixed key like in:

Cmajor:

Verse: C-Am-F-G
Bridge: Am-G, Am-G
Chorus: F-G-Am

How is this analysed?

1) Cmajor song, C root always
2) Cmajor song, C root, then A root in eolian, then F root in lydian mode?

I mean, when Bridge comes, is that a VIm-V or a new Im-V after a key change?

I guess my mistake is that C root and C key does not mean the same

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I would say that the song is in the key of C the whole time. I would only analyze the bridge as a modulation if you were not playing in Aeolian, which would have to be Dorian given the major chord a whole step below. However, if I were analyzing for a school assignment I would probably put both or justify why you chose one or the other in the written portion. If I were to analyze it as being in different modes I would probably say that the bridge and chorus are actually in Aeolian. With the chorus I would say this because I'm guessing it feels resolved when you arrive on the Am. Analysis is open to interpretation though so if you feel the F is root it may be. Also, the roman numeral you suggested for the G in the bridge (V) would actually be called VII or bVII in Aeolian: Im-VII, which could also be written: i-VII or i-bVII, where lower case roman numerals represent minor.

Root is usually used when talking about a chord, which represents the 1, even if 1 is not in the bass. The term tonic is usually used to refer to the 1st degree of the key you are in. If you choose to analyze each section as having its own mode, then I would say the tonic changes with each section.

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C key means tonality, C root is a C chord in the root position (with lowest note C). This can though have different interpretations in different countries, C root might also mean a chord (like F or Am) that has a C as lowest note.

There are always many ways to analyse, different ways to see it that are correct. I would say depending on the size/length of the bridge I would analyse as a Key change or not. Also if inside the section you have different chords that build a relation around the new I, then analyzing/concluding a key change would be more correct.

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The analysis is:

Verse: I VIm IV V Bridge: VIm V VIm V Chorus: IV V VIm

When you focus on the root and fifth of the VIm, you are creating a minor tonality. when you focus on the root and fifth of the I, you create a major tonality. The other chords are cadences that reinforce the tonality. A cadence may reinforce either the I or the VIm tonality.

The bridge is in the same key, not a different key, because V can resolve to VIm just as it resolves to Imaj.

When the song reinforces Am, then you say that part of the song is in Am (Aeolian). when the song reinforces Cmaj, then you say the song is in C major (or more specifically, the "key of the moment" is C major).

The roots of the chords don't tell you the tonality or key of the song, they just tell you where the chord starts.

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    I would just like to point out a couple things here: You need to do more than just "focus" on a root and fifth to create tonality. Also, not all other chords are cadences. Cadences are punctuation in musical phrasing. Saying every chord is a cadence is, like, typing, a. sentence. like. this. Cadences do not always reinforce tonality - sometimes doing exactly the opposite! Also, traditionally when using roman-numeral analysis, the numerals are lowercase for minor chords such that the Verse progression would look like: I-vi-IV-V. Just an FYI :) – jjmusicnotes Aug 27 '13 at 23:51
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    @jjmusicnotes: I've seen all manner of Roman numeral notation. I decided on a particular style for myself because it makes other notation unambiguous (I use lower roman numeral for key tones in the major scale.) Regarding your other comments, of course not all chords are cadences, but the point here is not terminology, but the concept of whether you are using chord progressions to establish tonality or not. – Michael Martinez Aug 28 '13 at 16:51
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    agreed that the point here is not terminology, however, terminology should be used correctly in addressing the intended point. Regarding roman numerals, there are many types of numeral notation - not all of them correct. What I put forth is the scholarly accepted convention of roman numeral notation. Made-up or esoteric systems may be helpful on an individual basis, but given the nature of this site, it is important to keep at least national interests in mind. – jjmusicnotes Aug 28 '13 at 18:08

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