What is the name of this this chord?

It is unequivocally a half-diminished seventh chord.

Why "half" diminished? The (fully) diminished seventh chord has two diminished intervals the diminished fifth and diminished seventh above the root. By comparison the half-diminished seventh chord has only one diminished interval the fifth. Its seventh is a minor seventh. So, one diminished interval is half of two diminished intervals.

Of the diatonic seventh chords it gets the Roman numeral (RNA) label viiø7 in major keys or iiø7 in minor keys. My feeling is the minor iiø7 is the more common usage, and that chord acts as a type of subdominant.

In D minor you would spell iiø7 as E G B♭ D. If it is inverted with the G in the bass it would be iiø6/5 and G B♭ D E.

Jazz/pop symbols come it two common forms that sort of match up with the Roman numeral inversions: root position is Em7♭5 an "E minor seven flat five" chord, or Gm6 a "G minor six" chord which is a G minor chord with an added major sixth above the bass.

Personally, the difference in labeling with RNA and jazz symbols annoys me. In RNA iiø identifies a subdominant chord regardless of inversion. But in jazz the m7♭5 is common for what would be a iiø7 subdominant in RNA, however m6 is often used for a tonic chord, like Gm6 in G minor. That's one function identified in RNA, but two different harmonic functions in jazz. I think of the m6 as a minor tonic chord with an added major sixth rather than an inverted half-diminished chord. That may be splitting hairs, or downright confusing, but I think worth considering when asking: "what is the name of this chord?"

'tis an E half diminished tetrad (chord with 4 notes). The difference between Diminshed and Half Diminshed tetrad is the use of seventh diminished (=sixth) or the seventh minor

Diminished tetrad: I, IIIb, Vb, VIIbb (E, G, Bb, Db)

Half Dimin. tetrad: I, IIIb, Vb, VIIb (E, G, Bb, D)

For tabs, the symbols used are a circle and a circle with a diagonal line

Diminished tetrad: E°

Half dimin. tetrad: EØ (i can't make this circle higher/smaller on smartphone, sorry)

EDIT: thanks, John Belzaguy

If those are the notes, then it's Em7♭5, in root position. E being the lowest, it is the root of the chord. It is often called a half diminished chord, which doesn't make too much sense, but that's what it gets called.

Those same notes, with G at the bottom, making that the root in root position, will be Gm6.