猫, ねこ, ネコ, neko
Which of these means "cat"? Is it all of them? Which situation do I use each of these in?
Firstly, rōmaji is not used to write Japanese. (The only Latin letters you find in Japanese text are abbreviations, like CD, OL, TPP, etc.)
Now, the question reduces to whether all of 猫, ねこ and ネコ can be used to say "cat". And the answer is "yes!".
Especially for plants and animals, it's often not so easy to decide which is the most natural choice in a given context (and the choice varies from person to person).
But in the case of "cat", the kanji 猫 has been a jōyō kanji since 1981, and is by a large margin the most common way of writing "cat" (data from the Balanced Corpus of Contemporary Written Japanese, via http://nlb.ninjal.ac.jp)
猫 6171 results (82%) ネコ 1340 results (18%) ねこ 54 results (1%)
For more info about the difference between kanji, katakana & hiragana for plants or animals see the following questions:
These are the most important rules.
However, names of animals and plants are a little bit special. In the context of natural science, katakana is used to write them. This is because it is easier to distinguish names of species from the other parts of sentences. http://ikimonotuusin.com/doc/289.htm
ネコ: in academic context. When you want to mention the species.
猫 or ねこ: in more general context. There is not much difference between 漢字 and ひらがな. ひらがな is softer.
Well, they are not that strict.