The term "high Mass" is rather vague. It more correctly refers to the Missa Solemnis or Solemn Mass, though it is sometimes used to mean a Missa Cantata or Sung Mass.
A low Mass is the most basic liturgy. It does not include music, sacred ministers other than the priest, or incense, and only two candles are lit on the altar.
A solemn Mass is a Mass in full ceremonial with a deacon and subdeacon assisting the priest, most parts chanted/sung, and the use of incense. Six candles are lit on the altar.
A sung Mass is the result of parishes wanting to have a solemn Mass but not being able to get the extra ministers required. Most parts of the Mass are chanted/sung, but there are no sacred ministers other than the priest. Incense may or may not be used. Four or six candles may be lit on the altar.
With the Novus Ordo, what is commonly done is sort of a variation on the sung Mass -- hymns are sung along with things like the Gloria and the Sanctus et Benedictus (but, for some reason, never the Credo), but most of the Mass itself is said. Incense is not used, and the priest is often the only sacred minister, though there may be a deacon.