It all depends on where you are staying. All the resorts around the island have several tavernas and restaurants. Skiathos town itself, is the place to be for a more varied nightlife. There are bars scattered all through the little streets and alongside the harbour a road known as 'Bar strip'. It has lots and lots of different bars, music bars and clubs all along this stretch, so you will have no problem in having places to choose from. Lots of these bars and clubs don't really get going til much later in the evening, but whenever we have walked past there are always people in there.
The buses run until about 1am (I think), so if you are outside the town, you may want to get a taxi for later on.
For earlier on in the evening around the harbour and up the main street, Papadamtis street you will find a good selection of places to eat and still more bars.
And as there never seems to be any rowdiness anywhere, I'm sure you will have a great time on a lovely island.
