Reported illnesses have ranged from people with mild symptoms to people being severely ill and dying.

Symptoms can include:

Fever
Cough
Shortness of breath

Typical Symptoms

COVID-19 typically causes flu-like symptoms including a fever and cough.

In some patients – particularly the elderly and others with other chronic health conditions – these symptoms can develop into pneumonia, with chest tightness, chest pain, and shortness of breath.

It seems to start with a fever, followed by a dry cough.

After a week, it can lead to shortness of breath, with about 20% of patients requiring hospital treatment.

Notably, the COVID-19 infection rarely seems to cause a runny nose, sneezing, or sore throat (these symptoms have been observed in only about 5% of patients). Sore throat, sneezing,and stuffy nose are most often signs of a cold.

80% of cases are mild
Based on all 72,314 cases of COVID-19 confirmed, suspected, and asymptomatic cases in China as of February 11, a paper by the Chinese CCDC released on February 17 and published in the Chinese Journal of Epidemiology has found that:

80.9% of infections are mild (with flu-like symptoms) and can recover at home.
13.8% are severe, developing severe diseases including pneumonia and shortness of breath.
4.7% as critical and can include: respiratory failure, septic shock, and multi-organ failure.
in about 2% of reported cases the virus is fatal.
Risk of death increases the older you are.
Relatively few cases are seen among children.

Pre-existing conditions

Pre-existing illnesses that put patients at higher risk:

cardiovascular disease
diabetes
chronic respiratory disease
hypertension
That said, some otherwise healthy people do seem to develop a severe form of pneumonia after being infected by the virus. The reason for this is being investigated as we try to learn more about this new virus.

Examples of possible development of symptoms (from actual cases)
A man in his 40s in Japan:

Day #1: malaise and muscle pain
later diagnosed with pneumonia
A man in his 60s in Japan:

Day #1: initial symptoms of low-grade fever and sore throat.
A man in his 40s in Japan:

Day #1: chills, sweating and malaise
Day #4: fever, muscle pain and cough
A woman in her 70s, in Japan:

Day #1: 38° fever for a few minutes
Day #2-3: went on a bus tour
Day #5: visited a medical institution
Day #6: showed symptoms of pneumonia.
A woman in her 40s, in Japan:

Day #1: low-grade fever
Day #2: 38° fever
Day #6: being treated at home.
A man in his 60s, in Japan:

Day #1: Cold
Day #6: Fever of 39° C. (102.2 F)
Day #8: Pneumonia
Another patient, in China with a history of type 2 diabetes and hypertension:

Jan. 22: Fever and cough
Feb. 5: Died
First death in the Philippines (a 44-year-old Chinese thought to have had other pre-existing health conditions):

Jan. 25: Fever, cough, and sore throat (hospitalized)
Developed severe pneumonia
Feb. 2: Died

Symptoms observed in hospitalized patients with COVID-19
Below we list the symptoms, with percentages representing the proportion of patients displaying that symptom, as observed in hospitalized patients tested and identified as having laboratory-confirmed COVID-19 infection. These findings refer to hospitalized patients, therefore generally representing serious or critical cases. The majority of cases of COVID-19 (about 80%) is mild.
Findings from the Wang et al study published on JAMA and based on 138 hospitalized patients

Common symptoms included:

Fever
98.6%
Fatigue
69.6%
Dry cough
59.4%
The median time observed:

from first symptom to → Dyspnea (Shortness of breath) = 5.0 days
from first symptom to → Hospital admission = 7.0 days
from first symptom to → ARDS (Acute Respiratory Distress Syndrome) = 8.0 days (when occurring)