The A-form barre is a little less tidy and comfortable to play than the E-form because its root is on the fifth string, making it less friendly to strum through. You will find that after a while you just know that you need to avoid playing that low E string. Muscle memory is a wonderful and powerful thing.
In most progressions, you will find that you will keep moving between E-form and A-form barre chords. The layout of the fretboard and the way music is structured mean that most chord shapes cluster in one region of the fingerboard in those two main forms. There are a couple of ways of grabbing that major shape: in Video 1 you'll see the three-finger method, which I find very uncomfortable and unreliable. Video 2 shows my preferred way of playing it, which is to form another mini-barre across strings 2-3-4 with my middle finger. That means I need to avoid playing the thin string because I'd be playing A6 if I let it ring out, not A. So the chord I wind up playing is a four-string chord, but it still consists of the required tones. It winds up being a 1-5-1-3 combination.
This is the A-Form family of barre chords, which come from the five main open A chord shapes. They are not quite as full sounding as the E-Form chords as they're only five strings wide. That thick E string can in fact come into play, as the note under the is the fifth of the chord.
However, playing the fifth as the lowest note of any chord is not the strongest inversion and will not sound right unless specifically called for. The root, or tonic, always sounds best as the low note unless you do want to use other inversions, which you probably won't for a while.
Video 1 and 2 | A-form - Major: This is the second most common way of playing a major barre chord. I find it a little harder to get a nice ringing sound using the A shape, especially the first version in video 1. I hate trying to get my three fingers into that little space, so I prefer the other way, which is in video 2, where I do the main barre with my index and a mini barre with my ring. I sacrifice the top string, but that's OK, I'm still playing all the required notes for a major chord. Also, it's best to not play that bass string as it's the fifth, not the root, of the chord. The root is found on the fifth string in this family of barre chords.
Video 3 | A-form - 7th: A lot easier to hold down than the plain old major. The orange note is an optional flat 7 note. In the video I do both. This is a good 7th shape to use, not as full sounding as the E-form as it's only 5 strings wide, but you should know all of the various ways of playing all chords. As with all of these A-form barre chords, the root of the chord is on the fifth string (second thickest), so it will sound a lot stronger if you don't play the sixth string.
Video 4 | A-form - Major 7th: This isn't the only way to barre a Major Seventh chord, but it certainly is the most common and the easiest on the fingers. That note on the third string is the 7, and the fact that it's jogged out of alignment with the other two makes it much easier to finger and get the chord ringing nicely. The bass note names the chord, but remember that the bass notes for all of these A-form barre chords is on the fifth string.
Video 5 | A-form - minor: This is a fairly easy minor chord shape, easily moved up the neck. You can see that it's like the E major shape, but it's moved up to the next string-set. If you compare this shape to the major shape (at the top), you can see that one note jogged out of alignment ... it's the flat 3, which is the note that makes it a minor chord. Remembering that minor chords are just like major chords but with a flat 3 will make it a lot easier
Video 6 | A-form - minor 7th: The same as the minor chord above, but with that Root on the third string gone, allowing the note under the barre to ring out. That note is a flat 7, so the chord becomes minor 7th flavored. It's all just logic. The orange note is an extra flat 7 that you can use or leave out. The chord doesn't change flavor one way or the other.