Should I fast consecutive six days in Shawwal, or should I fast on alternate days? And are these fasts considered fard, sunnat or nafl?
Nafl and sunnah are technically synonyms.
Nafl نَفْلٌ (singular نَافِلَة) in Arabic refers to something optional, or anything which is done as an addition of that what one should do.
While sunnah -when it comes to worships- refers to any kind of act, which is not prescribed as fard (obligatory) our Messenger (Peace be upon him) used to do in a more or less periodical way.
Generally in Arabic sunnah سُنَّة refers to something which is a kind of custom, it has been done this way for a long time either following a good or bad example see for example the meaning in Quran 15:13, 17:77, 18:55 etc.
Scholars may have different definitions to show to what level our Messenger used to do an optional worship. For example sunnah muakkada (emphasized sunnah) or raghibah (something asked for) may refer to a kind of worship our Messenger apparently never or rarely missed or asked or remained his sahaba to perform.
Fasting six days from shawwal is mostly considered a sunnah or nafl fast. It is allowed to fast them consecutively, but some scholars don't recommend it.
For details look at my answer on this post.