Corks work with cider in 2 ways:
1) You can use regular wine corks & bottles if you bottle still (NOT carbonated.)
2) You can use mushroom corks (or the plastic champagne stoppers) if you want to carbonate, but you have to also use champagne bottles, as regular wine bottles cannot withstand the pressure, and of course the wire hoods to secure the corks/stoppers.
The easiest way to have a sparkling (carbonated) cider is to use beer bottles. Either the type with crown caps, or the swing-tops; both are rated to take the pressure & are easily sealed. You can also keg, but if you're asking about bottling, odds are you don't already have a keg setup.