Your Tiny Tower is very tiny to start with. You have to build new floors to turn it into a real tower.
Building a New Floor
Scroll to the top of your tower and you will see a billboard that says Build Floor and has a number of coins below it. If you have that many floors, you can tap that billboard to add a new floor to the tower. This floor starts out as an Empty Floor.
Adding a Residence or Business
Once you have an Empty Floor in your tower, you can tap it, and you will be prompted to choose from Residential, Recreation, Service, Food, Creative, or Retail. Once you tap on one of these, construction will begin.
Construction takes a certain amount of time to complete. The higher the floor, the longer the construction takes. You can tap on the construction floor and spend Tower Bux to hurry the construction. Sometimes a construction VIP will appear in the elevator. If you take him or her to a floor under construction, it will subtract 3 hours from the construction time.
Your New Apartment or Business
After construction is done, the floor will turn into an apartment if you chose residential or a business if you chose a type of business. For apartments, the apartment name and decor are automatically chosen for you at random. Similarly, for businesses, the type of business is automatically chosen for you at random, based on the type of business you chose to build there. You can't control the type of business that is chosen aside from making it retail, food, creative, recreation, or service. See the list of businesses to see the specific types of businesses there are.