Oh god, so ill-defined. Not going be geologically, because I'd look at it very differently then.
North America - Mainland and minor outlying islands of the US, Canada, Mexico, Bermuda, and Greenland. Normally, I wouldn't consider Central America or the Caribbean part of either South or North America, but if pressed I would throw them here.
South America - Mainland and minor outlying islands of the Americas south of Panama/Colombia border (Darien Gap).
Europe - all of the European peninsula, most islands in the Mediterranean (including Malta and Cyprus), several islands in the north Atlantic (including the Azores and Iceland, but not Madeira or the Canaries), and the entirety of Turkey, Armenia, Georgia, and Azerbaijan. Eastern border gets cut off somewhere in Kazakhstan and Russia.
Africa - west and south of the Sinai peninsula; including most major Atlantic islands from Madeira south, some southern Mediterranean islands, Socotra, the Seychelles, Maritius/Reunion and other minor islands in that area.
Asia - everything on the mainland from (inclusive) Pakistan, Afghanistan, and Turkmenistan north and west that doesn't fall under Europe. Any Indian Ocean islands not already mentioned, the broader definitions of Nusantara, Japan, Papua New Guinea, and possibly the Solomon Islands.
Australia - usually just the continent proper. If I use Oceania instead, throw in the entirety of the Pacific islands not already mentioned, plus Papua New Guinea and the Solomons.
Antarctica - most things south of what I've already mentioned.
I decidedly do not consider the Middle East south of Turkey/Armenia/Azerbaijan/Turkmenistan, east of the Suez, and west of Afghanistan/Pakistan to be a part of Asia (or Europe, for that matter). It's just kind of it's own thing. I wouldn't call it a continent, but it wouldn't fit into my normal interpretation of any continent, either.