Numbers in the dots are suggested fingerings. Empty strings are muted or not played [most likely muted!].
x = mute, or don't play [mute!]. Numbers in brackets are optional tones which will require figuring out fingerings. Also, some of these chords are technically slash chords. Slash chords have a different tone in the bass than the root. In this set, Dm is specifically Dm/A, since the bass tone shown is an A. Yet, we rarely see it written as a Dm/A [though we typically let this open string ring for Dm].
The F also has an A ringing in the bass, but it can still be used when we see F. Another option for F, is to fret the 3 fret on the 5 string. This will make it F/C, which is often written as F/C. You can still use F/C when you see F [typically]. And, we can bar it [just the 1 and 2 strings - I'm not a fan]; or, all of it [only when necessary!].
The following is the Roman Numeric scheme for the chords in C Major and the 'fixed position' cycle of chord forms. For the numbering, C, F, and G are the I, IV, and V chords; Dm, Em, and Am are the ii, iii, and vi; B0 is the vii0 [know this!]. The cycle of forms are in blue. For more on this, see Chord Cycles.
As we would expect, the forms line up with the chord name for most of the chords. And, we can see that the E and A forms are repeated.
7 = Maj7 = Major 7 = 7th that is 11 half steps away from root.
7 = 7th is 10 half steps away from root.
If in a Major chord, it is a Dominant 7 [just 7]. If in a minor chord, it is a minor 7 [m7 or min7 or -7].
m7b5 = minor 7, flat 5 = half diminished 7 
The C scale is all natural tones. Notice that all of the tones in the chords are in the scale. For the tab above, the numbers below the tone names are first the traditional music theory numbering, and then the numerical chromatic equivalent.
As with all Major key centers, there are 7 tones in the scale (key). A chord can be built from every tone in the scale, thus giving us 7 chords. Chords are built by selecting a root and adding every other tone [see EON] until there are three tones [for triads].
A triad is a three tone chord built every other 'note'. 7 chords are 4 tone chords, taking the every other note process one tone further.
In a Major key, there are 3 Major triads [I IV V], 3 minor triads [ii iii vi], and 1 diminished triad [viio]. Upper case Roman Numeral means Major, lower case means minor, and lower case with a degree sign means diminished.
In a Major key, when chords include the 7's, the I and IV become Major 7's [IMaj7 IVMaj7], the V becomes a Dominant 7 [V7], the ii, iii, and vi become minor 7's [iim7 iiim7 vim7], & the viio becomes a minor 7 flat 5 [viim7b5 - also known as half diminished]. 7th chords are taking the EON principle until we have 4 tones.
The vii0 triad can be written as paralling to a minor triad and that would make it a minor flat 5. If it is a 7 chord, paralleled to minor7, this could be called minor 7 flat 5. Minor 7 flat 5 is also called half-diminished 7 and that is indicated as the degree sign with a slash through it. We see the min7b5 written more often than the degree sign with a slash. Just be aware that this type of chord has two ways to be written. They are the same thing.
1. We first get all of the chords in our hands. We take our time, fretting with confidence.
2. We then play the triadic chord scale ascending [and descending].
We play chords as we do scales [alphabetically]. We can use any beat count [4 is shown below] and any motor hand technique [strumming, fingerpicking, or arpeggio picking].
3. And then we play 7th chord scales after playing the triads with ease.
4. Next, we play progressions and/or songs that are in the key.
We keep in mind that the voicings shown in this lesson are but one way to play any particular type of chord. See the CAGED cluster for more options. Different styles favor certain voicings, and in a way are defined by the types that are used.
5. And, as always, we write progressions.
This process of arranging chords in our own order, and/or common sequences, is the genesis for writing our own songs. Start this process from the beginning.
We aren't limited to just the chords in a key for writing. Any chord can go to any chord. For understanding how other chords relate to a key center, check out Harmonic Map.