<p>What is the best way to track changes in a database table?</p>

<p>Imagine you got an application in which users (in the context of the application not DB users ) are able to change data which are store in some database table. What's the best way to track a history of all changes, so that you can show which user at what time change which data how?</p>

You've got a few issues here that don't relate well to each other.
At the basic database level you can track changes by having a separate table that gets an entry added to it via triggers on INSERT/UPDATE/DELETE statements. Thats the general way of tracking changes to a database table.
The other thing you want is to know which
user
made the change. Generally your triggers wouldn't know this. I'm assuming that if you want to know which user changed a piece of data then its possible that multiple users could change the same data.
There is no right way to do this, you'll probably want to have a separate table that your application code will insert a record into whenever a user updates some data in the other table, including user, timestamp and id of the changed record.
Make sure to use a transaction so you don't end up with cases where update gets done without the insert, or if you do the opposite order you don't end up with insert without the update.