Start systemd service after specific service?

In the .service file under the [Unit] section:

[Unit]
Description=My Website
After=syslog.target network.target mongodb.service

The important part is the mongodb.service

The manpage describes it however due to formatting it's not as clear on first sight

systemd.unit - well formatted

systemd.unit - not so well formatted

After= dependency is only effective when service including After= and service included by After= are both scheduled to start as part of your boot up.

Ex:

a.service
[Unit]
After=b.service

This way, if both a.service and b.service are enabled, then systemd will order b.service after a.service.

If I am not misunderstanding, what you are asking is how to start b.service when a.service starts even though b.service is not enabled.

The directive for this is Wants= or Requires= under [Unit].

website.service
[Unit]
Wants=mongodb.service
After=mongodb.service

The difference between Wants= and Requires= is that with Requires=, a failure to start b.service will cause the startup of a.service to fail, whereas with Wants=, a.service will start even if b.service fails. This is explained in detail on the man page of .unit.