Every year the University of Cambridge undergraduate course in mathematics welcomes students with a range of different qualifications. The individual Colleges handle all applications, and typical offers across colleges are all broadly the same. You should check individual College websites for Collegespecific requirements.
Offers
The majority of applicants will be taking either Alevels, or the IB, and the typical offer below refers to these. Applicants who are studying for other qualifications will receive an offer based on the kind of qualifications they are studying for. The Scottish Advanced Highers, for example, or the Cambridge PreU, are equally acceptable. If you are from overseas, the Cambridge Admissions Office provides a very useful guide of international qualifications.
All typical offers are also based on STEP (Sixth Term Examination Paper), which is an additional examination taken in the summer at the end of June, and is the same for all applicants. STEP papers do not require knowledge beyond A level Mathematics and Further Mathematics. Support programmes are available for STEP and for Further Mathematics.
Typical offers require:
 A Level: A*A*A (Mathematics, Further Mathematics + another subject) + 1,1 in STEP 2 and 3
 IB: 4041 points, with 776 at Higher Level + 1,1 in STEP 2 and 3
 Offers are also routinely made based on other qualifications
What we look for in a candidate
In selecting mathematics undergraduates, we seek three qualities:

Mathematical knowledge and skills
Students should have suitable background knowledge in mathematics, on track to start our course. Students must also have good general skills in mathematics to be able to apply their knowledge in solving problems. 
Mathematical ability and talent
Students should have the ability to think mathematically. This includes having the creativity to make connections between different mathematical ideas, and the flexibility to understand new concepts quickly and use them to solve challenging problems. 
Motivation for studying mathematics
Students should have the commitment to study mathematics at university level, and be ready to work hard and persevere. Considerable interest in and enthusiasm for the subject are important, and can be expressed in many ways.
We look for evidence of these qualities using many sources of information. These include existing and predicted grades, school reference, personal statement, and what we learn at time of interview. We do not simply grade and sum these sources separately. Instead we use all available information together, taking individual context into account, to form a picture of each applicant.
STEP is an important part of our conditional offer; it enables us to compare applicants directly, and addresses all three qualities above to some extent. All Colleges use STEP for mathematics, and a typical offer is grades 1 in both papers 2 and 3. However, only about twothirds of our places are filled by those who achieved these grades. For the remaining onethird of our places, Colleges review the full applications (now icluding STEP marks and scripts) of those who missed their STEP grades.
What subjects? Which modules?
Students often worry about what they should study. The best advice, applicable to all, is to do as much mathematics as possible. If you are taking Alevels, we recommend that you take Mathematics and Further Mathematics. If you are applying for Mathematics with Physics, most colleges will ask for Alevel Physics. Beyond that, we suggest that you study what you enjoy most and feel you can do well. If you have a choice of modules, you should try to take as much pure mathematics and mechanics as possible, in preference to discrete mathematics and statistics (from the point of view of our course).
We recognise, though, that for most of you there will be little or no choice, and some schools may not teach Further Mathematics.
You should not worry: all applications from talented and enthusiastic mathematicians are welcome, and suitable allowance is always made both in the interview and in the conditional offer, to take into account the background of individual applicants.
Support available for STEP and Further Mathematics
Do not worry if your school is not able to provide much help with STEP. There are many STEP preparation resources available to help and support you, including a new free STEP Support Programme developed by the Faculty of Mathematics.
If your school/college does not offer Further Mathematics A Level, the Further Mathematics Support Programme may be able to help you. The FMSP also offers an online Further Mathematics course for mature and homeschooled students.