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Can someone explain how I would use the word but in Japanese?

The word "but" does not really exist in Japanese as it does in English. You need to learn how to do compound sentences sentences in general. Technically, there are two kinds:
1. "and" relation
2. "but" relation
There is no "simple" way to explain it. Go here for the full monty:
http://www.guidetojapanese.org/learn/grammar/compound

the word "Ga" (that kana you got there, sorry, my Japanese text pack is not letting me type kana for some reason) would work as a "but" in the case of something like this:

Kutsu wa haite imasu ga, kutsushita wa haite imasen

basically translating to " I have shoes on, but I do not have socks on", "ga" in this case is "but", but "ga" can also mean (taken from Jisho.org):

1: indicates sentence subject (occasionally object);
2: indicates possessive (esp. in literary expressions);
3: but; however; still; and.

Hope that helps

I can't really say about the meaning actually, but I could help with my examples^^

A : "〇〇さんはいらっしゃいますか？"
B : "はい。　失礼です"が"、どちら様ですか？" = "Yes. I'm sorry, but may I have your name?"

the word "but" can be said other ways as well. For example, "～けど" or "でも".
・この服も欲しい"けど"、あの服も欲しい。/ この服も欲しい。でも、あの服も欲しい。
・彼女の気持ちも分かる"けど"、私は納得できない。/ 彼女の気持ちも分かるよ。でも、私は納得できない。

A : "００７のSky Fall面白かったね！！"
B : "面白かった～～!!!　でも、Casino Loyalの方が私はすきかな。"


We tend to use that way "でも" very often in conversation, but better not use it in social situation such as a conversation with clients or somebody in authority. However, the word "けど" is kind of flexible, but try not to use to your boss who is not close to you.
Hopefully this helps for your question and to make it clear^^

Hello Matt,
But has many translations but the most commonly used is "でも" (demo).
<I can't read the other answers cause there are many of them!>
Ex: A: Are you tired?
B: Yeah, but it's alright.
=> A: あなた は つかれ て いるか (Anata wa tsuka rete iru ka)
B: はい、でもだいじょぶです。(Hai, demo daijobudesu)
It can also mean "Yeah, I am still fine'/'I am going to be okay'

I think others did a better explanation than me since I am not a native speaker! This was a good question though...
Do your best がんばってください ^_^