"I learn English."
ならいます means to learn, yes and べんきょう します means to study. べんきょう just on it's own is a noun meaning study. And ならい is a verb derived noun created with the verb stem of ならう - ならand its base 2 - い. No fart jokes, please!!
Is "study" different from "learn"? I am not an english native speaker, so they are both translated to a same concept in my native language. Sorry for this out of topic question.
"Study" refers to the actions you take to "learn" things. Example sentence: "I study on Duolingo to learn Japanese."
Studying is the activity, Learning the result: I studied for a week and I learned greetings. I studied all day, but I did not learn anything. (English is not my native language either, there could be mistakes...)
The way I see it, to "study" means to actively seek knowledge, by reading books or going to school for example. To "learn" is a more general concept, it would refer to any knowledge you acquire, willingly or not.
I would say that studying is a mix of learning and intensely revising what you have learnt for memorisation/to retain what you have learnt. Study is often something that is done by yourself (although you can study in groups with varying degrees of success - my best friend and I could never study together because we would get distracted looking up funny words in thesaurus' and talking etc) and which requires personal discipline and dedication. Learning can be done by yourself or in a group - I would say that it is a process of familiarising yourself with new skills, knowledge and techniques etc - you can learn new things and add to or improve the skills and knowledge that you already have. Both learning and studying describe ongoing processes but studying is usually going over things you have learnt or in other words the process of making sure that you remember and keep current the things that you have learnt. Hope this makes sense.
You can learn without studying but you can't study without learning (I hope)
Yes it is different. To learn is to know new things. To study is reading, researching, memorizing, etc. all with the purpose of learning.
Because eigo is the direct object of narau ie. the thing that you are learning.
は is a topic marker, you shouldn't use it unless you want to change the topic. Proper translation for 英語はならいます would be something like "Speaking about English, I'm learning it". It may help to always put a comma after は to understand its importance.
Apparently manabu is more like self study/learning and narau is more for learning from someone.
It's definitely learn - check the kanji 学びます、学校、学者 - manabimasu (to learn), gakkou (school), gakusha (someone who is learned or in other words a scholar). - just editing this as it was in answer to your original comment which you've since edited. Your comment now seems more in line with my own experience of these two verbs. When you learn something from someone it is ni narau - an uncommon usage of ni where it means from eg. haha ni narau - I learn from (or by) my Mum. Both the from and the by sound odd/unnatural in English but they are helpful for explaining what the ni is there for in the Japanese even if you don't include that in your translation. Another instance where ni is used like this is with morau (to receive) tomodachi ni tsukutte moraimashita - I had my friend make it for me - but literally I received the making of it/something from my friend - or something like that.
Yeah i realized my original comment maybe didn't specify the difference enough. It's hard trying to explain basic japanese briefly on the Internet lol.
When do we use desu instead of masu? And how would we say "I am learning X" instead of "I learn X"
-masu is a verb ending - it is not used by itself like desu is. -masu will always be attached to the end of a verb. In answer to your second question the simple answer is ｘ を 習っています - I am learning x, x を 習います - I learn/will learn x.