Warning: too much information alert.
A chord is a group of notes played simultaneously. A chord can be defined in different ways in different music theory systems, but we'll stick to the diatonic system used by the Dead and virtually everyone else in the Western world.
A triad is a group of three notes. This is generally going to be a major, minor, augmented or diminished chord consisting of three notes chosen from a scale, each of which is assigned a numerical value starting on the "root" note of the scale. "Root" = "1." The C major scale:
C D E F G A B
1 2 3 4 5 6 7
The interval- the distance between the notes describing their relationship to one another- is the crucial unit to understand. From one fret to the next on any string is a half-step. Two frets is a whole step. Any scale is a collection of intervals. The major scale consists of seven intervals and the are arranged like this, starting from the root:
whole whole half whole whole whole half
That means that the distance from the root to the second is a whole step, from the second to the third is a whole step, from the third to the fourth is a half step, from the fourth to the fifth is a whole step, from the fifth to the sixth is a whole step, from the sixth to the major seventh is a whole step, and then from the major seventh to the octave is a half step. Play the major scale up the 5th string from C to see the relationships. This set of relationships is the same for every major scale, no matter what note you start on.
A good musician doesn't really play notes, they play intervals. Intervals have emotional weight to them and pull the music in different directions. The Dead played the intervals, the spaces between the notes.
A major triad chord is made of the root, major third and fifth intervals: C E G is the C major chord. The intervals are a major third (C to E) and a minor third (E to G). Make sure you see those relationships. A minor triad chord is made of the root, minor third and fifth: D F A. Make sure that you see those intervals, too. Major and minor chords occur naturally within all keys.
Diminished chords and augmented chords are rarely used in rock music, but the Dead used them frequently (e.g., Ship of Fools, The Music Never Stopped). A diminished chord is made up of two minor third intervals- in the key of C: B D F (root, minor third, minor third) and an augmented chord is made up of two major third intervals (say, G D C). Of note, diminished chords occur naturally in all keys but augmented chords do not occur naturally in any key. That's why they always sound dissonant. I can't offhand recall any Dead tunes with an augmented chord. The minor seventh flat five (min7b5) chord is very close to the diminished chord (Terrapin Station has them in several places).
An arpeggio is the notes of a chord played individually, often in sequence but not necessarily. Fingerpicking is a form of playing arpeggios. Arpeggios are often played as lines- bass players do this all the time and soloing instruments, too. Listen to Ben Webster or Clifford Brown; you can hear them outlining the harmony of the song in arpeggios and thus you can actually hear the chord sequence in their solos- if there was no accompaniment you could still follow the chords. Jerry did this, too, although he was also a strongly modal player (there are other threads on that).
Someone above confused chord additions (9ths, 11ths, 13ths, etc.) with arpeggios. That's incorrect. Those additions are called extensions. Sometimes your will come across the term "altered" chord which generally means playing a chord with the or ot two notes changed a half step higher or lower:
Hope this helps!