You can trigger a build manually as described in the Travis CI documentation.
If you want to trigger a build programmatically, use travis-ping to forge a post-receive hook and trigger a build:
git clone https://github.com/jpmckinney/travis-ping.git
cd travis-ping
bundle
require 'travis-ping'
Travis::Ping.ping 'github-user', 'travis-token', 'repository-owner-name', 'repository-name'