I do my fresh egg pasta fairly regularly, without a machine. It is quite similar to the one Kadesma posted, but let me copy my recipe from my previous post here, a bit more detailed.
Here is what I do...
the main ingredients are flour, eggs and water.
Use one medium egg to every 100g of flour (about 3,5oz).
Keep a small jug of water and additional flour nearby and handy.
Wash your hands well.
Pile the flour on a flat board, making a well in the middle.
Crack the eggs into the well, then carefully mix in the flour from the top edge into the egg mixture, gradually blending everything together. (At this point some people stir the eggs with folk before starting to blend in the flour, but I usually don't bother with that) Don't worry if egg escapes and oozes outside, just push it back up with the outside edge of your hand and keep blending it with the flour.
Once the eggs and flour are blended, start kneading vigorously and evenly, for at least 15 minutes until the dough is completely smooth and elastic.
You have to feel the dough while you are kneading. If the dough is too sticky and gooey, add a little flour. If the dough is too dry and hard, add some water. At the beginning of the kneading the dough may feel quite unmanageable, but that is also normal, keep on working it and adjusting the texture with water or flour, sooner or later it will become much "tamer"!!
Form a ball and cover it with a wet cloth, let it rest for about 30 minutes.
Dust well the board with flour.
Then flatten the dough with a rolling pin evenly, dividing the dough in a few pieces as needed, to about 1-1,5mm thickness.
At this point, you can also make it into lasagna/cannelloni sheets, or cut into thin strips to make tagliatelle. It is (at least for me) almost impossible to stretch the dough into a regular shape (rectangle or circle), if the odd pieces are left at the edges, that is quite normal and you can also use that, cut them up in anyway possible and cook it and enjoy that with your choice of sugo. (they also have an official name in Italian "maltagliate", or badly cut, but widely used anyway, after all, they taste good just as well!)
When you cook them, it takes much less than dried pasta to be cooked well. Boil plenty of water in a large pot, do not overcrowd the water with the pasta. Scoop them up as they float onto the surface.
Another tip I discovered pretty recently. Mix the egg first (so the white and yolk are already blended together) before you place it inside the well of flour. It makes much less mess and easier to handle!!