Screen printing is a printing techique used to push ink through a mesh stencil onto textiles. Each colour in your design requires a different stencil (or screen) to be made. Once the screens are made, all colours are lined up (or register) to one another for printing.
Even after the introduction of digital garment printing, screen printing remains the best process for printing textiles.
Silkscreen is by far and away the best method of printing T-shirts. It's the industry standard for printing on T-shirts. Other methods, such as Direct-to-Garment printing, are best left for small runs i.e. one-offs or when the quantity is so low that doesn't warrant the cost of the screens.
For information about screen printing, here is a reference Wikipedia
Why my t-shirts doesn't feel like those of bigger brands?
There are many factors to why your final product don't feel like it belongs on a shopping rack.
1. Fabric Quality
There are many different kind of fabric available. Some fabric are made of 100% cotton and some are 50/50 cotton polyester mix. Different fabric gives the product different finishes.
2. Printing and Ink
Experience of the printers plays a huge part. Additional colours helps to pop up the colours for a 4-colour CMYK screen print. There are many different printing methods for different kinds of ink too.
3. Design / Cutting / Print
The most important part of the process, the cutting, print and material. It's a learning curve, keep trying until you get it right.