I am trying to learn new words but pronunciation is difficult. I try to learn new words, but by the next morning I have forgotten what they were and I have to start over again. How many new words should I learn every day so that I can remember them all?
In other words, as many as you are able to learn. If you can learn 10 a day, you'll be doing great.
The more you read (stories, newspaper articles, textbooks, not EFL textbooks, but textbooks in fields that you know something about and are interested in), the more vocabulary you'll learn and the faster you'll learn it.
Don't try to memorize vocabulary lists: that's pointless because it doesn't work. To be able to remember vocabulary, you have to recycle it, which means that you have to read those words in meaningful and interesting contexts as often as possible.
My European immigrant father-in-law learned English vocabulary by reading the newspaper (with a dictionary) every day. When I was studying French last century, I read an anthropology textbook written in French. I had to use the dictionary a lot at the beginning, but by the time I'd reached page 100, I'd learned (not merely memorized) all the important vocabulary and could rapidly read the rest of the book (another 150 pages) without needing the dictionary more than once or twice every ten pages. I love to read anthropology books, so it wasn't a chore.
This is a never ending task in every language. Even though I was raised speaking English, I still learn new English words from time to time. There are many websites, screen-savers, etc. whose purpose is to teach (or remind) born-and-raised English speakers of vocabulary words.
That said, I think this is really a personal choice. Some people pick up certain types of vocabulary words very quickly, others spend more time making each word part of their vocabulary. From my experience, I would say just go as fast (and learn as many new words each day) as you can comfortably and consistently remember - there's no point in wasting time on a word that you won't remember.
One trick that has helped me when learning vocabulary in English or other languages is to use "Flash Cards". Write the word on one side of a piece of paper, and the definition or sample usage on the other. Then you can flip through them every day and quiz yourself to make sure you still know each word. (It can also be fun to have someone else quiz you - take turns asking the other person a word.) You can also flip the cards over and read the definition (don't put a usage on the back in this case), and "guess" which word fits that definition.
There is no such universal magic amount of words to learn.
It is up to you how fast and how well you learn new words. There are many good techniques that can help, for example you could learn only a few words per day, and repeat them over time.
Flash cards are a practical application of this.
If over time you feel you forget the words you knew only a few days back, then you are learning them too fast, and you should slow down.
Also, there are some words harder to learn than others. It is usually related to the mother tongue of the student: the more similar both words, the easier you'll learn them (never forget false friends: similar words with different meaning for each language).
There's not a right answer for that question. It depends on you. The more you study, read and listen and practice conversation, the more you will learn.
A good habit to keep when you are learning vocabulary is to keep a word list of the words you have learned, and update it regularly. There are great sites for that, such as www.vocabulary.com.
English is not my first language, so I use that site to keep track of the words I learn, and there are great quizzes and usage examples, that help me reinforce my learning.
I also use the word frequency lists from wiktionary: http://en.wiktionary.org/wiki/Wiktionary:Frequency_lists#English
I usually go through the list, and find the words I don't know, and add it to my vocabulary list. It's a great way to learn the easier words that you haven't learned yet, that you should know first.
Although this question already has several good answers, I think they can be improved.
How many should I learn per day?
The literature on second language acquisition is divided over exactly how many words you need to have good competency, and the estimates vary wildly over tens of thousands. From experience, my French and German were good enough to read newspapers and novels when I knew about 2000 words - but many, many more are needed to be able to read literature well. The question is therefore, how quickly do you want to learn? If you want to reach a reasonable productive competency in two years, this will equate to learning just 3 words a day. You should be able to remember those! If you want better compentency faster, you need to learn more.
The key is to only do as many as you can revise later: it's no good learning a word and just forgetting it (see below).
When I was starting Russian and Greek, I could only do about ten words at a time before reaching the "my-brain-is-melting" point and finding that no more words would go in in a single session. Chinese was worse. But I can now learn upwards of several hundred French or German words in one go: I promise that it will get easier to learn more words with times as you get used to converting between spelling and sounds, and as you get used to the combinations of sounds used in a particular language.
How can I remember words?
It is absolutely true that using words in a real-world context helps you to remember. However, the above assertion that you cannot learn from vocabulary lists is simply untrue - many education systems including China rely heavily on memorising vocabulary lists. I prefer to learn from vocabulary lists simply because I don't like looking up words when I'm reading, and I love the satisfaction of seeing a word I've learned and being able to understand it - this is what make learning languages fun for me (this of course varies)! However, I can also tell you from experience:
You should absolutely expect to forget words after you've learned them. They need to be learned on multiple occasions before they will go in. I recommend reading through them or writing them out as first contact, and then testing yourself on them orally or in writing until they've all gone in. And repeat a little later. And then on a different day.
I think it's really important to make sure you know the pronunciation when you're learning a word - and saying out loud to yourself or even in your head when you're drilling it will help you to remember it. To find the prounciation, look up how to read IPA, and find the phonetic transcription in any good dictionary. There are also websites with pronunciations. (I would recommend this even for classical languages - I have found that it really helps with learning.)
Of course - you might choose not to use vocabulary lists at all, and just look up words when you come across them: they will eventually go in as well!