This is in the Bass clef, no key signature, and called F (Major I assume) but why isn't it C or A?
At first I thought that this would be a duplicate of What determines a chord's name?, but I don't believe it is. That thread discusses chord quality, not actually determining the root!
So: we call these three-note chords "triads," and we name them by what we call the "root" (which is a particular note) and "quality" (which is an adjective like "major" or 'minor"). You can learn about quality in the above link, but as for the root, it comes down to a process that we often call stacking the thirds. Although the chord you listed does have a C and an A in it in addition to the F, we call it an F triad because F is the bottom-most pitch when those three pitches are stacked in thirds.
But before we explain that, we have to first talk about measuring intervals. To measure intervals, simply count the note names from the first to the last. And when we measure intervals in music, we use what we call inclusive counting, meaning we count the first and last note name. So if we're measuring the interval from G to B, we count G ("one"), A ("two"), and B("three"). G to B is thus a third.
Now, back to stacking thirds. Consider the three voicings of this chord shown below:
In the first example, notice that we have a third between A and C (A is one, B is two, C is three), but a fourth between C and F (C is one, D is two, E is three, F is four). These are thus not stacked thirds, so the bottom-most pitch is not the root.
In the second example, we have the same fourth from C to F. Once again these are not stacked thirds, so the bottom-most pitch is not the root.
The last example, however, has a third from F to A and a third from A to C. Since these are stacked thirds, the bottom-most pitch, F, is the root. We thus call this an F triad.
As a shorthand, you'll notice that the stacked-third version (what we call a "root position" chord) has all of the pitches on either a line or a space. So to put a triad in root position faster, simply put all the pitches right next to each other so that all pitches are on a line or space. In the first example above, the F is the odd pitch out, since it's the only pitch on a line while the other two are on spaces. So let's move the F down an octave so that it too is on a space, and voila! We have our root-position triad.
This all leads us to an important distinction in the world of music: the difference between root and bass. The bass is the lowest-sounding pitch. But the root is the lowest pitch when you stack thirds, even though it's not necessarily the lowest-sounding pitch. It's a tricky distinction, but a vital one.
Lastly, you're correct that this is F major, but determining chord quality takes a few more steps. I'll refer you again to What determines a chord's name? if you're interested!
As Richard has explained, the way to determine the root is to move all the notes around (up or down an octave) until they all stack up in thirds, and then the bottom note is the root. In the chord in your example, move the C up an octave to middle C, and they will be in thirds, with the bottom note an F. So, it's an F chord.
The reason that we do it this way (rather than always naming the chord by its bottom note) is that inverted chords, or chords where the root isn't on the bottom, are essentially the same chord as the chord in root position, or not inverted. Displacing a note by an octave doesn't change the quality of it as much as moving notes by some other interval. Try playing your chord, then try moving the C up an octave and playing it again. They will sound similar. Now, try moving the C up one step to D. It is now a D minor chord, and will sound quite different from any inversion of the F chord.
A chord with the third on the bottom (the middle note if the chord is in root position) is said to be in first inversion. A chord with the fifth on the bottom (the top note if the chord is in root position) is said to be in second inversion. Your chord is an F chord in second inversion.