One party will normally need to inform the court that there has been a settlement. Both parties will then be contractually obliged to honour the agreed terms of the settlement agreement.
If a mediation settles, the parties will have a signed settlement agreement. This is a legally binding document and the parties will be under a legal obligation to comply with it.
One party will also have to let the court know that there has been a settlement to end the court process. This may be carried out by way of a consent order (or "Tomlin order"). As part of the mediation process, the mediator will confirm with the parties how the parties will end any other processes.