The DADGAD tuning is also known as Celtic tuning and hence often used for playing Celtic music on the guitar. It can also be interested in lots of other styles if you like to experience.
Essentially, you tune to a Dsus4 chord (that is what you hear if you play all string open). Since the tones are close to many D chords, the tuning is extra suitable for playing in the D key.
The name says it all, but just to be extra clear, to get this alternate tuning you should tune your guitar according to this: D A D G A D. Don't increase the string tension when you re-tune.
Including an octave perspective, the tuning is written D2-A2-D3-G3-A3-D4, meaning that the lowest string is a D note on the second octave, the second lowest string is a A note on the second octave and so on. Standard tuning reference: E2-A2-D3-G3-B3-E4.
Chords (closed and open)
The chords presented are in the key of D and most of them are movable. For example, 3332XX would be F major and 3331XX would be F minor.
You can choose to include the two highest strings or not. In most cases you get a more characteristic sound related to this tuning if you do.
If you have problem to get enough room for the fingers you could remove the finger from the 5th string and mute if with an adjacent finger, for example A major as 7X76XX, which is formally named A(no5). There are many possibilities for voicings, with the high strings played opens, for example the lovely sounding Aadd11 as 770600. See below for pdf chord chart ("Essential Chords in DADGAD Tuning ebook") for many more voicings, progressions etc.
The alternate D chord could also be played without the finger on the fourth string, in that case the correct chord name is D(no3).
More chords in DADGAD tuning
Other chords in this tuning, including open chords:
D: 12 12 12 11 X X
D6: 004220 / 024400
D7: 000234 / XX0234
Dm9: 00098 10
F: XXX 10 12 10
Asus4: 777777 / X 0 2 2 0 0
C: X3203X / X32032
Chord progressions with DADGAD tuning
Some examples of progressions using this alternate tuning:
D - Gadd2 - Add4 - D
Bm7 - Gadd2 - Add4 - D
Em7 - D - Gadd2 - Aad4
Em7 - F#madd - Gadd2 - D
Finally, a tip is to lock at the possibilities to play octaves with the 1st and 4th string. Octaves will be found on the same fret starting with the D octave on open strings.
If you want to go more into depth of this particular guitar tuning, see the Essential Chords in DADGAD Tuning ebook with over 300 chord diagrams.