I’m about to begin a two year stint as an English as a Foreign Language teacher. Now, this question will be different for everyone, because duh, people are individuals, and what’s difficult will often depend on your native language. Still, I started thinking about this earlier today, and now I’m curious: if English isn’t your first language, what was the most difficult thing about it when you studied it?
Pronounciation, mainly. Both its irregularity (very irregular compared to German) and those phonemes that are not part of the German repertoire.
Old joke: A German couple in London read a newspaper headline in the entertainment section: “Cats pronounced success.” Remarks the husband: See, that’s just like them.
Grammar wasn’t a problem (compared to English French and Italian grammar are definitely more difficult for me, and Russian grammar is the stuff of nightmares).
You know, I can’t really name anything. Spelling, pronounciation, grammar, the words themselves, it all came pretty naturally to me.
I’ll answer on behalf of my wife, who is Korean. I’d say she’s 90% fluent in English as long as the conversations do not get too technical. I think she’d say the following three things were the hardest for her:
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Pronunciation. Actually, her pronunciation is not bad at all. She just carries that “Asian ESL” accent that will be with her the rest of her life.
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Word order. She tells me that I am the best husband “whole in the world.” Things like that. Also the use of articles. Korean doesn’t have them and they can be confusing.
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Idiomatic phrases. She wants to be able to casually drop them in to conversation and not mess them up. Boy she studies that idiom and usage book a lot! (“Birds with feathers flock at the same time.” No, no, honey, close but that’s not it.)
My boyfriend’s Serbian, and he has trouble with articles too. He tends to use articles in sentences that don’t need them (he says, “As a kids, we used to…”) and leave them out for sentences that do (“I was drinking bottle of water”).
He also asks questions with words in the wrong order. “Why you did that?” “Why I can’t have one?”
He also says that he had trouble with sentences like “While I was watching TV, he was playing video games” because he had trouble with the verb tenses. However, I haven’t noticed him making errors like this, so I’d say he did catch on in the end.
I also need to add that I find it easier to understand him than most people who speak English as a first language. He spells well and speaks clearly. 
I work with lots of ESL speakers, primarily in 2 groups: Indians and Russians.
One Russian in particular has problems with that/which/, etc (relative pronouns?). She generally uses “which one”: “We need to talk about the project which one is overdue.”
The Indians have trouble pronouncing V’s/W’s.
I learned English too young to have any real problems, but my mum is Chinese and a lot like rinni’s boyfriend - putting articles where they’re not needed and omitting them from where they are, having trouble mixing tenses, and putting words in the wrong order. These, incidentally, are the main problems I have when learning French. She also tends to pronounce some words wrong and when corrected can’t tell the difference between her version and the correct version.
The irregular and highly fickle pronunciation of vowels probably gives lots of ESLers trouble.
This sounds familiar! (Bulgarian and Serbian are, I think, pretty closely related.) My counterpart - the Bulgarian English teacher I work with - always says “the” in front of every city. Like, “We’re going to the Sofia next week.”
Pronunciation, and sometimes I forget that if you put “did” in the sentence, the verb that follows is not supposed to be in the past tense (as in “Did you took out the trash?” instead of “Did you take out the trash?”).
Verb tenses themselves (and articles), other than the above thing, are a piece of cake, when you come from a language that has a lot of them (Spanish).
In/on/at are also a bitch to me, since in Spanish we only have “en”. Most of the time I get it correct, but if I’m speaking too fast I confuse them.
I’ll always carry my foreign accent when speaking English.
Boy from Mars is Italian, he says grammar was the most difficult thing, because if you are from a latin language (i.e. French/Italian) background, the sentence construction compared to English is entirely different. This is specifically in what order subject-verb-object goes, which I think others have also mentioned. Plurals also tend to flow from Italian to English - “I need to get my hairs cut” as the word is capelli (plural) in Italian.
My extremely intelligent and well educated Russian former boss would also make this type of error from time to time.
Another phrase he would habitually use was something like “We must complete this project as fast as only we can” when he meant “…as fast as we can”.