The phrase 'on again' means to start something that had stopped or to resume an activity. Some synonyms for 'on again' could be 'commence once more,' 'restart,' 'recommence,' 'resume,' 'pick up again,' 'continue,' 'revive,' 're-establish,' 'rekindle' and 'reinstate.' For example, when a computer falls asleep, you have to press the power button to turn it on again. You can also say you have to restart the computer. Similarly, if a movie stopped playing in the middle, you would have to hit play to start it on again. Additionally, the phrase could refer to a couple on a break who decide to try their relationship on again.