There are TONS of pasta dough recipes out there, the most basic made with just water and flour. Aside from all these recipes, there are different types of flour that can be used, too - semolina, all purpose, whole wheat, etc. Then of course you can mix and match the flours but that's a whole other story.
I prefer to make an enriched dough. An enriched dough is one that has anything other than water, flour and salt (and yeast, depending on the type of dough.) My recipe of choice is Anne Burrell's pasta dough recipe. It is enriched with both eggs and olive oil.
Here's the recipe. Any added notes are in blue text!
1 pound all-purpose flour
4 whole eggs, plus 1 yolk (use local, organic eggs if possible for this recipe)
1/4 cup extra-virgin olive oil
1 to 2 tablespoons water or more if needed
Put the flour on a clean dry work surface. Make a hole (this is also called a well) in the center of the flour pile that is about 8 inches wide (bigger is definitely better here). Crack all of the eggs and the yolk into the hole and add the olive oil, salt and water.
Using a fork beat the eggs together with the olive oil, water and salt. Using the fork, begin to incorporate the flour into the egg mixture; be careful not to break the sides of the well or the egg mixture will run all over your board and you will have a big mess! Also, don't worry about the lumps. When enough flour has been incorporated into the egg mixture that it will not run all over the place when the sides of the well are broken, begin to use your hands to really get everything well combined. If the mixture is tight and dry, wet your hands and begin kneading with wet hands. When the mixture has really come together to a homogeneous mixture, THEN you can start kneading. (At this point - before kneading- wrap the dough in plastic wrap and let rest for 10 minutes at room temperature. This will make the kneading process easier!)
When kneading it is VERY important to put your body weight into it, get on top of the dough to really stretch it and not to tear the dough. Using the heels of your palms, roll the dough to create a very smooooooth, supple dough. When done the dough should look VERY smooth and feel almost velvety. Kneading will usually take from 8 to 10 minutes for an experienced kneader and 10 to 15 for an inexperienced kneader. Put your body weight into it, you need to knead! This is where the perfect, toothsome texture of your pasta is formed. Get in there and have fun!
When the pasta has been kneaded to the perfect consistency, wrap it in plastic and let rest for at least 1 hour. If using immediately do not refrigerate. (This dough will hold in the fridge for a few days. When you're ready to use the dough, remove from the fridge and let it sit at room temperature for 30 minutes to an hour.)
Roll and cut the pasta into desired shape. (A traditional hand-powered pasta maker is typically capable of making linguine or spaghetti. You can also purchase an extruder attachment for a Kitchen Aid to make tubular shaped pasta. Toss the freshly shaped pasta in all purpose flour or semolina flour to keep them from sticking. Before cooking the pasta, shake off the excess flour. I find that gently tossing the pasta in a strainer works well. Pasta can be shaped and then frozen, but I prefer to make the pasta on the day that I cook it. You can also make the pasta earlier in the day and then just keep it in the fridge until you're ready to use it. Just toss the pasta occasionally with your hands to make sure it doesn't stick together!)
How smooth and supple! (Calm down Anne!)