what do you call a t-shirt making machine?

A commecial one? It totally depends on the type, they have different names. One style has multiple stiff 'octapus' arms that rotate, each pressing does a different color so the ink filling bottles are above each presser.

Home printing of t-shirts would require a standard computer printer and those transfer sheets.

Your best bet is to call a t-shirt printing place in your phone book and just ask them.