Japanese text: ねこはコンピューターをみています。
Romaji: Neko wa KONPYUUTAA wo miteimasu.Translation: The cat is looking at the computer.
The are many styles of romaji. One style writes out the Japanese character that would be used, and another just uses a macron (the line above a vowel to denote that it is a long pronunciation) for long characters. Another example:
高校 = こうこう = koukou = kōkō = high school
In this example, the word is pronounced with a long O sound for both O's. However, this is just an example of two different styles of romaji. There are more and they get pretty technical. I use a mix of different kinds because it's the way that I learn it best, and I suggest that you do the same. As for now, just work with what you're comfortable with. As you learn hiragana and katakana, you'll realize that you'll run into minor issues, such as both ず and づ both having the same sound, "zu". For these situations, I suggest finding a way that works for you, such as writing ず as "zu", and づ as "du" or "dzu".
One last thing to note. In the first example with the cat looking at a computer, you may have realized that romaji is written in all caps for katakana. I am unsure if this is a certain style of romaji also, but it helps denote which words are written in hiragana and which are written in katakana. For those of you who like to karaoke to Japanese songs, you may already know this fact if you look up song lyrics and read the romanization.
In short, romaji is just a way for Japanese to be written with the Latin alphabet. You may even call it an unofficial alphabet of the Japanese language. Pretty much all English speakers learning Japanese depend on romaji to help them read and understand Japanese when they first start off. However, after you've learned all your hiragana and katakana, you'll rarely be using romaji anymore (and you shouldn't depend too much on it either, since you want to push yourself to actually learn hiragana and katakana). This is why I think it's best to write romaji in a way that works most efficient for you. Develop your own style to it, but also keep in mind the other styles too so if you come across it, you can recognize and read it.