This gets asked a lot, maybe this would be useful as a sticky.
How mathematical is a particular economics course?
The answer depends on the university and the best rule of thumb this is what their A-level requirements are. If they recommend doing further maths then it's a pretty mathematical course. If they want A level maths it's pretty standard, if they want A grade probably quite mathematical, if they want B maybe less so. If they just want GCSE maths then probably they minimise the maths content and they may well put those without A-level maths on to maths classes in their first year to catch up the essentials.
Is a BSc more mathematical than a BA?
Sometimes people say "if its a BA it's not mathematical, if it's a BSc it is mathematical", this is a red herring as it's a convention of the university: Cambridge, Oxford, Durham etc call theirs BA this does not mean that they are maths-light. At some universities they offer two economics courses: one that's designed for students who want to progress to MSc which has a number of compulsory maths and stats courses especially in year 2 and 3 and another that doesn't have compulsory courses and is broad choice. Generally in this case they will use BSc and BA titles to distinguish them and in this case the BSc is likely to be the one that has the compulsory maths courses (although there is nothing to stop a BA student from taking some of them). But you have to look at the individual university and the curriculum to check that, you can't make inferences directly from the BA/BSc title.
How much maths do you need to study economics?
The answer is the further you take economics the more you are likely to be exposed to difficult maths. The core material on an undergraduate programme is not massively mathematical, it's not really different from A-level maths. By this I mean the maths used in the standard microeconomics and macroeconomics textbooks that are used to teach undergraduate programmes anywhere in the world (there are only a handful of recognised texts and the core curriculum is quite similar).
MSc courses are a big step up in the level of maths and then if you go on to PhD depending on your area of focus you may be engaging in very difficult maths. However note that some universities (traditionally the most prestigious, Cambridge, LSE, Warwick etc) will load up their respective curriculums at each level with a higher level of maths than you need to follow the core micro and macro at that level, because they are gearing people up for further study. If you do an undergraduate on one of these more mathematical courses then moving to a Masters will be less of a jump up as you will be familiar with the material. And again at Masters, some of the more mathematical economics departments will load up their MScs with even higher maths to prepare their students for entry to PhD programmes.
Generally - if you just want to do an undergraduate degree in economics you don't have to be a maths genius to follow standard undergraduate level micro and macro courses. If you are serious about wanting to do an MSc then it will help you down the line to do a more mathematical programme at undergrad.
So what kind of maths is involved?
The big difference between A-level maths and the maths you use in economics is the (narrower) breadth of topics: A-level (especially further) has a lot wider breadth of maths with mechanics, trigonometry etc that doesn't really come in to economics. The main mathematical areas in economics are calculus (mostly differentiation with some integration, largely around optimising functions), matrices (and how to do matrix algebra), and statistics, which will flow naturally from A-level type statistics on hypothesis testing, confidence intervals, normal distribution etc in to regressions and the broader topic of econometrics.
For an undergraduate degree, some basics that you will use a lot are:
General algebra, rearranging equations (GCSE stuff), using rules of indices and rules of logs as these are often in the algebra.
Geometric series - these appear a lot so worth being familiar with the standard formulae, also compounding (compound interest, compound growth etc).
Differentiation - you need to be at a point where using rules like product rule, quotient rule, chain rule are natural, like doing basic arithmetic. You do a lot of finding minimum and maximum points and will be taught how to do these optimisation problems when there is a constraint to the function (not sure if that's covered at A-level).
Integration - comes up less often, but if you can differentiate it's not that difficult to learn this.
Statistics - using E and V operators and doing algebra with the sigma summation sign. This often trips a lot of students up as it can feel a bit awkward and also looks psychologically daunting as the equations have sigma all over the place.
Knowing how to use normal distribution, t-distribution, F-distribution, how to read the tables, how to find critical values and test hypothesis at different levels of significance.
Matrices - finding inverses, transposes, ranks, determinants and doing matrix algebra (ie where you don't have the matrix written out but just have letters to represent each matrix.
IMO the heart of economics particularly micro is differentiation so if you get on top of this you will basically be fine. The challenge in moving from A-level type differentiation to what you will see on an economics course is at A-level you generally differentiate functions which have the numbers already written in. In economics sometimes it is a bit more abstract as you will be working with things like v=(p,x(p)) which means v is a function of p and x, and x itself is a function of p, so to differentiate it you will need to use a chain rule in there because there's a function of a function. The further you progress (particularly at MSc) the more the problems move away from having numbers in and towards general forms so there are a lot of alphas and betas in there - you are using the same rules of maths, but you are doing it in a different way from what you did at A-level. That can feel unnatural at first so takes a bit of time to settle but once you crack it it's fine.
What books can I get to prepare?
The best starter books for an A-level (or someone who hasn't done A-level maths) student would be:
Ian Jacques Mathematics for Economics and Business
Geoff Renshaw Maths for Economics
Jacques in particular is a fantastic book for self-teaching and you can work through this in your summer before 1st year at uni and be well prepared even if you weren't confident on the maths before as he really goes from beginners level.
For a stronger A-level student who is doing further maths I would recommend:
Alpha Chiang and Kevin Wainwright Fundamental Methods of Mathematical Economics
This is a classic book which in various versions has been a stable of economics courses for decades. It goes through the basics, but it presents in a more formal mathematical way than the two above and so can be a bit dense for an A-level student. This is quite comprehensive and includes a lot of topics that go right through an undergraduate course and also MSc.
Pemberton and Rau Mathematics for Economics
Similar to Chiang and Wainwright, fairly concise but comprehensive. This would be another good one for a confident A-level Further Maths student that is eager to get a bit ahead of themselves on the curriculum to get a good sense of what they will do at university.
Simon and Blume Mathematics for Economists
This is another classic and is a good reference for the maths on a lot of topics that are used in economics but can be a bit heavy going so is more useful for postgraduates, PhDs etc.