"I had a party the day before yesterday."
Okay you want the history on how these pronunciations were originated.
The origin was 「をとつひ」. をと（遠 or 彼方） meant "far away", つ was the old form of の, and ひ（日）is day. So it meant "A day which is far away."
This became おとつい and eventually おととい in pronunciation.
Kanji 一昨日 was introduced when Chinese characters started to be used in Japan. Since then, the Chinese reading (On'yomi) was introduced as いっさくじつ.
ます is just the polite non-past verb ending that would attach to the verb stem. It wouldn't be used by itself.
します is the ます form of the verb する "to do"
You need a "do" verb here because パーティー isn't a verb, it is a noun.
パーティーをします lit. "to do a party" translated more naturally to "to have a party"