What would you call this chord? The guitar is in standard tuning. No capo.
It's basically the open C fingering slid up 2 frets, while keeping the open strings open.
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Even though Shevliaskovic's suggestion that this could be an Em9 (no 5) chord is theoretically correct, it is hardly ever used that way. If that voicing were used as an Em9 chord there would be no reason to indicate that the low E string should be muted (note the cross on the low E string). With the note D in the bass, the Em9 chord would have its seventh in the bass, which is not impossible, but which only happens if there's an appropriate reason for it in a specific context. It definitely wouldn't be the standard way to voice an Em9 chord.
You can check if in the piece where you found that chord it is really used as an Em9 chord, by simply replacing it by a standard Em voicing and checking if it sounds right. I suppose it doesn't.
I've seen this chord being used very often, and I've used it myself very often, and in all cases it is used as a chord with the root D. One of the many examples where you can see and hear it is in The A Team by Ed Sheeran (2:43). Here it is definitely used as a D chord (actually E because of the capo on the 2nd fret).
So, what should this chord be called then? It is used as a D major chord with some added colors. There is no 5th, but that's pretty common because the 5th usually doesn't add much color anyway. It has an added 9th, a standard tension in popular music. And it also has an added 4th, which is usually frowned upon when used with a major chord. However, on the guitar in this very voicing with the open G string it simply sounds good in certain contexts. If the guitar were tuned differently, people probably wouldn't play that chord.
If you really want a chord symbol (which doesn't make so much sense because this voicing is specifically used on the guitar, not so much on other instruments), use
D add(4,9) (no 5). But as I said, the specific voicing is important here, so if you want to communicate that chord/voicing to others, you should better specify how to play that chord instead of using a chord symbol which does not indicate a voicing.
It can function as an Em9 (ii,vi) a Dmaj11 (I) or a D11 (V). The key and the chords before and after in the progression will define its function. If it comes after a CM in the key of G, it would possible to call it a D11 (V). Since it doesn't have the leading tone that makes it a true dom chord (c) it could also be used as a pivot chord to the key of D the chord would function as a Dmaj11 (I). If it comes after a CM in the key of G, it's likely functioning as an em9 Chord (vi). It could also be functioning as an em9 (ii) In the key of D.
tldr; Its function depends on what key you're in.