A triad is the most used set of notes making a chord. Obviously it's 3! However, there must be a word/ term for a set of 4 notes - as in maj 7th, or min 6th chord, and maybe a different one for a set of 5, as in a 9th chord. What is the term?
The name is indeed tetrad, as pointed out in a comment by Shevliaskovic. However, this term is not very common. Standard tetrads built in thirds are almost always referred to as seventh-chords. The problem is that there are two common four-part chords (yes, that's actually a very good name!) that are no seventh chords, because they contain a major sixth instead: X6, Xm6 (replace X with any root you like). They can be referred to as sixth chords, but traditionally they would be called added tone chords, or, more specifically, added sixth chords.
In sum, for chords arranged in thirds, the most common names are seventh-chords, ninth-chords, eleventh-chords, etc. Chords consisting of a triad and other added tones (not in thirds) can be referred to as "added tone chords", or if you want to be more specific, as "added Xth chord", where X specifies the added scale degree (such as a sixth). For chords with more general structures I think that "N-part chord" (N=4,5,...) is the clearest and most common name.
I believe that the terms "pentad" (etc.) are not at all common in music theory. There exist terms like pentachord and hexachord, but they do not necessarily refer to chords but to series of five or six notes.
the word triad is not specific to music and simply means a group of three. In general the series is, monad, diad, triad, tetrad, pentad, hexad, heptad etc.
Check out this list of polygon names (http://mathcentral.uregina.ca/qq/database/QQ.09.96/rosa1.html) I'm reasonably sure you can just replace the letters "agon" with "ad" should you ever need more than seven notes in a chord.
There's no generally used name beyond dyad (2) and triad (3) (some would dispute that a dyad deserves the label 'chord', but that's a different question.)
Note that although we have a chord naming system (mostly) based on a 'pile of thirds' - triads, sevenths, ninths etc. - many three-note chords are not triads, many 4-note ones are not sevenths... And there are many possible chords that don't fall into our 'chord symbol' naming system at all.