We still don't fully understand how the new coronavirus spreads, but we're learning more every day.
While animals are believed to be the original source, the virus spread is now from person to person (human-to-human transmission). There is not enough epidemiological information at this time to determine how easily this virus spreads between people, but it is currently estimated that, on average, one infected person will infect between two and three other people.
The virus seems to be transmitted mainly via small respiratory droplets through sneezing, coughing, or when people interact with each other for some time in close proximity (usually less than one metre). These droplets can then be inhaled, or they can land on surfaces that others may come into contact with, who can then get infected when they touch their nose, mouth or eyes. The virus can survive on different surfaces from several hours (copper, cardboard) up to a few days (plastic and stainless steel). However, the amount of viable virus declines over time and may not always be present in sufficient numbers to cause infection.
The incubation period for COVID-19 (i.e. the time between exposure to the virus and onset of symptoms) is currently estimated to bet between one and 14 days.
We know that the virus can be transmitted when people who are infected show symptoms such as coughing. There is also some evidence suggesting that transmission can occur from a person that is infected even two days before showing symptoms; however, uncertainties remain about the effect of transmission by  non-symptomatic persons.
If you want more in-depth information the an interesting article about how are people being infected with COVID-19 can be found on the website of livescience.
Source: ECDC (European Centre for Disease Prevention and Control)