Screen printing or silk screening is where you make a stencil on a screen and use it on a print press, pushing ink through it onto the shirt.
Heat press or transfer is where you print a design on transfer paper and heat press it onto the shirt.
Heat pressing is easier and cheaper to get into so a lot of people do it, but most mainstream shirt companies use screen printing. Screen printing also has a lot more potential as far as what you can print it on.
Ink jet printing is starting to surface, but it is very expensive to get into.