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E or E7 chords seem to work very well in the key of C, particularly at the end of a line/verse before a powerful chorus for instance.
As one example I end a verse on Am - E... hang on the E then jump into a chorus line F - C - G - Am.
Clearly Em is the minor 3rd chord but musically, what is E or E7 and why do they work so well?
It's a secondary dominant. As such, it can be used to move to its own tonic, in this case, A minor. Other secondary dominants in key C are B7, leading to Em, and A7, leading to Dm. Both the Dm and Em are diatonic chords in key C.
There are also C7, which is non-diatonic, to lead to F, and D7, leading to the dominant G. So, the dominant of the dominant, or V/V.
Your E moving to F could be called a deceptive cadence - not going where expected.
It seems that secondary dominants don't have to go directly to their 'tonics', but they often do.
As mentioned in (the original version of) Tim's answer, it's indeed a secondary dominant, but I'd like to add that the resolution E7 to F is a specific resolution called deceptive resolution, and it occurs a lot, in almost any style of music. Note that F major and A minor share two out of three chord tones, so it's not so surprising that this type of resolution also works. A different, less used type of deceptive resolution in C major is the cadence G7 to Am.
E (or E7) is the dominant of the relative minor of C major. Therefore very much 'home territory'. Not diatonic, but pretty darn close!
And E leads well to F because all three notes move up by a semitone. Like leading notes do.
I have a theory of 'honorary diatonic' chords that include (in C major context) E and E7, B♭, C7 and Fm (when used in the cliche 'Saints' C, C7, F, Fm, G7, C ending sequence). Maybe also any secondary dominant that doesn't progress to an established modulation and any ♭5 substitution for one. My 'honorary diatonics' are chromatic chords that DON'T lose their contact with the home tonality. They don't need to be described in relation to some supposed change of key centre, or as a 'borrowing'. E7 - F doesn't have to be explained away as a frustrated cadence to A minor, it can have a legitimate identity as III7 - IV in C major.
Once we admit that a B♭ or E7 chord 'belongs' in C major and doesn't have to be described as related or 'borrowed' from some other key, analysis gets much simpler!
As mentioned before, E or E7 in the key of C usually acts as a secondary dominant. It's the dominant of both A major and A minor chords. It's often used in progression like E-A-D-G-C (perhaps with some sevenths or minors instead of major chords here.) The other important case is E-a; in this case, it's the dominant of the relative minor. Moving between a key and its relative minor is very common. Check out "Delilah" by Tom Jones. (In reverse, one could have a Folia Progression based on A minor so one would get a-E-a-G-C-G-a-E-a.)
In a Cycle of Fifths (a.k.a. Circle of Fifths) progression in a major key, the viiº chord is often omitted so one gets (in C major), C-F-E-a-d-G, skipping the tritone progression and the diminished chord. As an aside, in a minor key (c-minor) the whole Cycle of Fifths is commonly used as the diminished chord occurs in a "nice" place: c-f-Bb-Eb-Ab-dº-G-c.
One thing to add to the secondary dominant/deceptive resolution thing is this: a dominant is a harmonically stable chord to end a phrase on (exluding the final ending.)
If the structure of the section is something like...
| ...Am E :||: F | C G | Am...
It may make more sense to describe the part before the double bar repeat a kind of incomplete, half cadence ending. After the repeat C G | Am... would seem more of a deceptive progression. If the E and F belong to separate sections, it doesn't make much sense identifying a deceptive cadence/progression there, because they are separated.
I put a double bar repeat in that example just to make clear I'm indicating separate song sections. You may not actually have repeats or sections like that in your song.
