I understand how HTTPS connections work, and I also understand they they require a certificate to know that the secure connection you have is with the right server (and not someone pretending to be the right server). My question is, how are these certificates secure? Why is it hard to fake them?
SSL is based on public/private key cryptography. This means that
A certificate is a public key with information about the owner attached to it. The owner can either be a person or a domain.
When you visit https://stackauth.com, the server will present its certificate to your browser. The browser checks that the name in the certificate matches the name of the website you requested and if the expire dates are valid.
Then the browser checks whether the certificate was signed by a certificate authority that the browser knows as trustworthy. Anyone who creates a faked certificate has the challenge to get it signed by a certification authority that is trusted by the browsers. There may be a chain of signatures from intermediate certificate authorities which needs to eventually lead to a trusted one.
If any of those checks fails, the browser will show you a warning and require confirmation. If all the checks are passed (or overridden by the user), the browser will use the public key from the certificate to encrypt some information that only the server can decrypt. This information is used to setup a secure session in a couple of more steps.
The main issues with this model is that the certification authorities are the weak link. There is a huge number of CA, that are directly or indirectly trusted by the common browsers. All of them can issue faked signatures, and this has allerady happened a number of times in the past (e. g. Verisign, Comodo, Diginotar).