I
Me, Myself
pronoun, の adjective
The kanji and the word are exactly the same. That means they share meanings as well.
我 is an old-fashioned pronoun and isn't commonly used outside of set phrases like 我を忘れる (to forget oneself or be utterly absorbed) or 我に返る (to come back to reality or return to one's senses).
Since this word is made up of a single kanji, it probably uses the kun'yomi reading. You did not learn this reading, so here's a mnemonic to help you remember:
It is I! Due to my ego, I love the Wii and no other gaming device. I love playing all the Wii-ware (われ) that's available, and everything else sucks.
お店が開くと、みんな我も我もとセール品を目掛けて走り出した。
When the store opened, everyone ran towards the sale items, each of them focused on themselves.
子供時代は、我を忘れて夢中で楽しめることがたくさんあった
During my childhood, there were many things I could enjoy wholeheartedly, forgetting myself.
我に返った時には、取り返しのつかないことになっていた。
When I came to my senses, it had become an irreversible situation.
我関せずって感じのアイツの態度、超嫌なんだけど。
I hate how he acts as if he doesn’t care.