What are techniques to make homemade pasta without a pasta machine?
Many pasta shapes can be made without a pasta machine--home cooks have done it for generations.   The only pastas that are really difficult to make without a special machine, I think, are the extruded pastas like buccatelli or macaroni.

One common method, which leads to a lot of different shapes, is to roll the pasta dough out into a sheet with a rolling pin.  It can then be cut into a variety of shapes, with a knife, pizza cutter, or even a cookie or biscuit cutter:


To make tagliatelle or linguine, you would first roll out a sheet.  Then, to make it easy to cut into strips, roll the sheet up, as if you were rolling a jelly roll.  Use a very sharp knife to cut it into ribbons.
Cut wider strips for lasagna noodles.
Cut into rectangles for homemade cannelloni.
Use a biscuit cutter to make circles, and you can fill them and fold them into tortellini.


Some more interesting shapes, like orecchiette, can be shaped by hand as well.  You create a small cylinder of dough, and then cut it into coins, then flatten these into the final shape.  The linked article has detailed instructions and pictures.

I have not tried to go into a huge amount of detail, because you are going to want to read specific recipe and technique articles, and watch videos of the techniques, all of which you can easily search for.  Each of the links above is to a recipe that has lots of helpful pictures and detail.