There are a few ways universities may accept this ‘proof’- you can have studied for a minimum period (determined by each university) in English, or an English-speaking country, or complete an international qualification such as the IB Diploma (in English!). However, if neither of these apply to your particular situation – perhaps you are completing a national curriculum in a non-English speaking country – you may need to submit test scores to show your level of English fluency is at the required level.
The most widely accepted test is the IELTS, which provides test-takers with a number from 0.5-9 which indicates their English proficiency.
Oxford, for instance, ask for an overall score of 7.0, with minimum 6.5/7.0 per component, depending on the course being applied to, while other universities might ask for an overall score of anything from 5.5-7.5 for undergraduate study.
Additional accepted qualifications are TOEFL, IGCSE English (Language), or English as part of a national curriculum, although you must ensure your particular course fulfils all requirements of each of the universities you are applying to. You will not have to provide your proof of English proficiency at the time of submitting your UCAS application, and so be sure to focus your time and effort on making that personal statement as wonderful as possible, but it is worth doing some cursory research ahead of the deadline so you are mentally prepared to study for a test should you need to.