This means postfix will only be used locally, so in order to be safe prevent outside users from contacting your mailer by configuring your firewall to block port 25 incoming.
Edit postfix' main.cf and set the following, I think this should be enough to get you up and running. Make sure to study the other settings too if they also apply to your situation.
# The inet_interfaces parameter specifies the network interface
# addresses that this mail system receives mail on. By default,
# the software claims all active interfaces on the machine. The
# parameter also controls delivery of mail to user@[ip.address].
#
# See also the proxy_interfaces parameter, for network addresses that
# are forwarded to us via a proxy or network address translator.
#
# Note: you need to stop/start Postfix when this parameter changes.
#
#inet_interfaces = all
#inet_interfaces = $myhostname
#inet_interfaces = $myhostname, localhost
inet_interfaces = localhost
# INTERNET HOST AND DOMAIN NAMES
#
# The myhostname parameter specifies the internet hostname of this
# mail system. The default is to use the fully-qualified domain name
# from gethostname(). $myhostname is used as a default value for many
# other configuration parameters.
#
#myhostname = host.domain.tld
#myhostname = virtual.domain.tld
# The mydomain parameter specifies the local internet domain name.
# The default is to use $myhostname minus the first component.
# $mydomain is used as a default value for many other configuration
# parameters.
#
#mydomain = domain.tld
# The relayhost parameter specifies the default host to send mail to
# when no entry is matched in the optional transport(5) table. When
# no relayhost is given, mail is routed directly to the destination.
#
# On an intranet, specify the organizational domain name. If your
# internal DNS uses no MX records, specify the name of the intranet
# gateway host instead.
#
# In the case of SMTP, specify a domain, host, host:port, [host]:port,
# [address] or [address]:port; the form [host] turns off MX lookups.
#
# If you're connected via UUCP, see also the default_transport parameter.
#
#relayhost = $mydomain
#relayhost = [gateway.my.domain]
#relayhost = [mailserver.isp.tld]
#relayhost = uucphost
#relayhost = [an.ip.add.ress]