Yes, the process is very different. US universities not only take into account your grades, but also your extracurriculars, your personality, your whole character basically. Also, you and your parents are expected to contribute towards your education, however Columbia offers good financial aid and meets 100% of your demonstrated need. For Columbia, the average award for international students who apply for and receive financial aid is $57,727. (Columbia is also need-aware, which means they take into account how much they have to give you into the admission decision).
For Columbia you'd typically need really good GCSEs and A-levels. (Mostly A*s and As)
You use something called the 'Common Application'. There you fill out details about your extracurriculars, honours, write your personal essay, write supplemental questions for the university, invite your teacher and counsellor recommendations. There are some guides for the Common App available online -- maybe check EducationUSA or the Fulbright UK-US website.
You also need to do the ACT or the SAT, you can Google when and where these are held (bear in mind it costs to take these). You can also buy books to help you with them. If you do the ACT, I would recommend trying to get a minimum of a 33 on your ACT (it's scored out of 36), as the range is 32-35 and the average is 34. Obviously it's not all about your ACT score, but a higher score definitely helps (for example - I got in with a 30, and the university's range was 33-35!)
You'll also need some good extracurriculars and honours to talk about on the common app. Good ECs generally include leadership positions such as Youth Parliament, leader/founder of a club/society at your school, etc., also sports, music, computer programming, politics, work, volunteering, NCS; there are loads of things that are considered a good extracurricular. You can google some examples of ECs (do not lie on your application!). Honours are typically academic honours, e.g. you received a national award for an essay, or a school award for getting the best grade in GCSE maths, or even something like Oxford UNIQ or the Cambridge Shadowing Scheme. Again, you can google examples of what they are.
For your personal essay, there are some guides online to help you with this.
You'll also need a counsellor and 2 teacher references. Your counsellor submits your transcript (all grades from official exams you've taken: GCSEs and A-levels/A-level predictions) and also a counsellor letter of recommendation and also a school profile (again, I'm sure you can find lots of info on this through google). Your 2 teacher references will also need to submit a letter of recommendation each. LoRs tend to be very different in the US, and I'm sure there are lots of help online.
You'll also need to fill out the CSS profile with all of your parents finances (its very detailed!) and whatever else Columbia requires.
Columbia is very hard to get into, and not everyone -- even the best students -- are able to get in. Please bear that in mind if you choose to only apply for Columbia. But my advice is to apply to a range of schools: there are loads of fantastic universities and liberal arts colleges (don't forget about these, they're fantastic) in the USA.
This short guide is not only for Columbia -- but it can be applied to lots of other unis and colleges. Good luck!