Translation:He is my older brother.
If you're talking about your older brother with someone outside of your immediate family, use 兄 (あに, ani). If you're talking about someone else's brother, use お兄さん (oniisan), the most polite version. If you're talking to your brother in japanese, にいさん (niisan) is the word to use. You can also use oniisan and niisan when talking to young men whose name you don't know. The same applies for other family-member-words as well, except you don't use お母さん and お父さん to mean middle aged woman/man. Use obasan and ojisan for that instead.
As far as i know, you usually refer to people by their name, and most of the time after the subject is established, pronouns arent needed. So like "マリアは姉です. きれいです。" is "Mary is my older sister. (She) is pretty." I generally would refrain from third person pronouns unless you dont know the person's name, right? Feel free to correct me if im wrong!
Just adding to Cheri's answer, あいつ is actually a pretty rude/condescending way to refer to someone. It's relatively common in casual speech, but generally only if the person you're referring to is well out of earshot.
As Cheri mentioned, names are preferred, but using かれ or かのじょ isn't considered rude, like あいつ is.
Just to be clear in English: "elder brother" means a brother who is older in age than the speaker or than their sibling(s). "Elderly" as an adjective means "very old". "He is her elder brother" means he is older than her (but they could be children or young people). "He is her elderly brother" means he is very old, and implies that he is also many years older than her. So "ani" means elder brother, not elderly!
More thoughts on the English use of elder:
In my experience, elder is an old-fashioned way of saying older. At least where I come from, it isn't very commonly used as an adjective. As a noun, it means someone older than you - as in "Children should respect their elders."
Elderly is also an old-fashioned word, but used commonly as a polite way of saying old.
An elderly brother wouldn't need to be many years older, just old - chances are the sibling in question is elderly as well!
兄 itself is already older brother (in fact, when you look up 兄貴 in the dictionary it refers to the former). The 貴 is an old-fashioned way of expressing politeness towards elders/people of higher status.
Fun fact: that's the same "ki" as in "ki-sama" 貴様. Once a highly honorific way to address someone, but nowadays closer to an insult.
Yes, it is somewhat redundant in most cases, but あに doesn't explicitly mean "MY older brother". The "MY" part is simply very heavily implied due to Japanese social dynamics. It can be used to refer to other people's older brothers, for example your cousin's older brother when talking to somone outside your family, so it could simply be that the speaker is making sure there isn't any misunderstanding.