Pure fresh semolina pasta with and without eggs
There are two pasta manufacturing techniques. First, pasta can be cut to shape using a blade or roller, and second, it can be extruded at high pressure through a bronze or teflon die. The first is what most people know of as fresh pasta, and is made using common flour and eggs. The second is made using semolina flour and water. If you are using a standard Atlas or Imperia roller, or cutting with a knife, use flour and eggs. It is the traditional technique and the semolina will not make a good pasta. If you have access to industrial pasta extrusion equipment, use semolina. 

Semolina is typically made from a different species of wheat, durum wheat, which has a very high gluten content, so it does not require eggs to bind it together. The egg white has very high protein content, which compensates for the lower protein in common flour. In McGee's, "On Food and Cooking" he explains how the extrusion process aids the formation of a firm pasta:


  The movement, pressure, and heat of extrusion change the structure of the dough by shearing the protein network apart, mixing it more intimately with starch granulas that have been partially gelated by the heat and pressure, and allowing broken protein bonds to re-form and stabilize the new network.


In "Essentials of Classic Italian Cooking", Marcella Hazan says:


  "[semolina] is the only suitable flour for industrially produced pasta, but I do not prefer it for home use. To begin with, it's consistency is often grainy, even when it is sold as pasta flour, and grainy semolina is frustrating to work with. Even when it is milled to the fine, silky texture you need, you must use a machine to roll it out; to try to do so with a rolling pin is to face a nearly hopeless struggle. My advice is to leave semolina flour to factories and to commercial pasta makes: At home use unbleached all-purpose flour.