How should I implement an auto-updater?

I think that "language agnostic" is going to be a limiting factor here. Applications come in so many shapes and sizes that there is no one-size-fits-all answer. I have implemented several auto-updaters in several languages, and no two were similar.

The most general philosophy is that the application checks with some home location (web address, web query, corporate network location, etc.) to either ask if it's version is current, or ask what the most current version is. If the answer calls for an update, that process will be different for each situation.

A popular alternative is to invite the home location to run a script when the application is initiated. The script can check the version, download updates if necessary, and ask for usage feedback, for example.

We can probably help better if you narrow the parameters.

UPDATE: The approach to "patching" also depends on the nature of the application, and there's a very wide diversity here. If you have a single executable file, for instance, then it's probably most practical to replace the executable. If your application has many files, you should look for ways to minimize the number of files replaced. If your application is highly customized or parameterized, you should strive to minimize the re-tailoring effort. If your application employs interpreted code (such as an Excel VBA application or MS Access MDB application), then you may be able to replace parts of the code. In a Java application you may only need to replace a JAR file, or even a subset of the JAR contents. You'll also need to have a way to recognize the current client version, and update it appropriately. I could go on and on, but I hope you see my point about diversity. This is one of those many times when the best answer usually starts with "Well, it depends ...!" That's why so many answers include "Please narrow the parameters."

Be sure to also consider the security implications of sucking down information about the update, as well as the update binaries themselves.

Do you trust the source of the download? You maybe phoning home to got your update, but what if there is a man in the middle who redirects to a malicious server. An HTTPS or similar secure connection will help, but double checking the bits that you eventually download by using a digital signature check is recommended.