知らない　 always 'I don't know' わかりません　could be 'I don't know', could be 'I don't understand'. It depends. e.g.) When is his birthday? ----> you can say '知らない' or 'わかりません' both are ok. But if you don't understand when a Japanese person talks to you, you should say 'わかりません'. In this case, '知りません' doesn't work.

知る（知っている) contains, besides that means 'acquainted with, be familiar with', the meanings of 分かる（分かっている）that imply understanding, comprehending, perceiving or identifying.

知らない　usually concerns with knowledge 分かりません　usually concerns with understanding

Hi, as two members said, the basic difference is "not to know" and "not to understan" but there are more differences. I found good explanations on internet and I'd like to share them with you. It explain about 知っている (= you are in the situation of knowing about something) and わかる. 知っている ・to know because of having the knowledge ・to know through your experience ・to know the meaning of the word/sentence but literally Ex.) この国の名前を知っていますか。-- 知りません。 (Do you know the name of this contry? -- I don't know.) あの女優が結婚したのを知りませんでした。(I didn't know that actress had gotten married.) これは誰の本ですか。-- 知りません。(Whose book is this? -- I don't know.) わかる ・to be able to distinguish/recognize ・to understand ・to know/understand the reason why ・to know/understand how to solve the problem ・to understand the whole/real meaning of something Ex.) ・この文章の意味が分かりません。(I don't know/get the meaning of this sentence.) ・あなたがなぜ泣いているのか、分かりません。(I don't know why you are crying.) ・分かりましたか。何か質問がありますか。(Do you understand? Do you have question(s)?) ・どちらが本物か分からない。(I don't distinguish/recognize which is authentic.) I hope my English isn't very bad and this helps you. Japanese people, teachers and advanced learners, please add more explanations.