〜ませんでした
didn't (polite past)
Verb ます-stem + ませんでした
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Add でした to ません and you have the polite past negative: 食べませんでした, "(I) didn't eat". It's long, but it's two pieces you already know.
The conversational alternative is the plain past negative plus です: 食べなかったです. Both are accepted here.
A common slip is 食べないでした. It sounds logical, but the past goes on the negative ending itself, not with でした added afterwards.
One thing it isn't for: "haven't done it yet". That uses まだ with the ている form (まだ食べていません), which comes later. ませんでした is for a finished stretch of time: 昨日は何も食べませんでした, "I didn't eat anything yesterday."
Register: Polite. 〜なかったです is the conversational alternative.
Examples
昨日はどこにも行きませんでした。
I didn't go anywhere yesterday.
今朝は朝ご飯を食べませんでした。
I didn't have breakfast this morning.
先週は雨が降りませんでした。
It didn't rain last week.
パーティーに田中さんは来ませんでした。
Mr Tanaka didn't come to the party.
昨日は勉強しませんでした。
I didn't study yesterday.