In which order does Ansible look for its configuration file, and which one wins?

ANSIBLE_CONFIG (environment variable) first, then ./ansible.cfg in the current directory, then ~/.ansible.cfg in the home directory, and only then /etc/ansible/ansible.cfg. The first file found is used.

* Ansible walks down this list and stops at the first file it finds. *

1. ANSIBLE_CONFIG: if this environment variable points to a file, that file overrides everything else.
2. ./ansible.cfg: a file in the directory where you run the ansible command. This is what makes per-project settings work.
3. ~/.ansible.cfg: the user's personal file in the home directory.
4. /etc/ansible/ansible.cfg: the global base file shipped with the package, used only if nothing else is found.
5. If even that is missing, Ansible uses its built-in defaults.

The more specific and more deliberate a source is, the higher it ranks: an explicit variable beats the project, the project beats the user, the user beats the system.

Tip: read it bottom-up as the order in which they were introduced (global, personal, project, variable); the search order is the reverse.

Go deeper:

• Ansible Configuration Settings: the configuration file (official docs) — the search order and the rule that the first file found wins.