The TOEFL is an American test, based on American English, so it's the one required for most American universities. I didn't realize it was also sometimes required in Europe.
It's developed and administered by the Educational Testing Service, a US nonprofit organization which also produces the SAT (for American high school students applying to a university), the GRE (for university students applying to graduate school), the LSAT (law school), MCAT (medical school), etc.http://www.ets.org/toefl/
The Cambridge exams are British tests, based on British English and administered by the University of Cambridge. I'm not familiar with the different ones, but there should be information on their website.http://www.cambridge.org/elt/
The IELTS seems to be partly administered by the British Council and to include a business English option as well as one for academic English.http://www.ielts.org/
In any country, some universities may accept more than one test, while others may require a single particular test. If you're considering a particular university, you should visit its website or write to its admissions office.