Great
Considerable, Important, Big Deal
adjective
した is the past tense form of する (which means "to do"). So, essentially this word is "did big." What is something that "did big"? It's something that's great or considerable.
The reading for the 大 part is actually the on'yomi reading you learned when learning this kanji. Since 大 has two on'yomi readings, here's a mnemonic to help you remember which one to use:
When something's great and important, you'd better wear a tie (たい) for it. No showing up without a tie to anything that's a big deal, okay?
大したものだ
quite a feat
大した女だ
she's a great woman, you're a great woman
大したことはない
not a big deal
大したことじゃないよ。
It's no big deal.
えっ！？コウイチが、メジャーリーグに入る？ それは、大したものだ！
What!? Koichi's entering the major league? That's a big deal!
このオレをふるなんて、大した女だったよ。
She had considerable nerve, dumping me.