The time before the symptoms of a viral infection appear is called the incubation period. During this time, viral genomes are replicating and the host is responding, producing cytokines such as interferon that can have global effects, leading to the classical symptoms of an acute infection (e.g., fever, malaise, aches, pains, and nausea). These symptoms are called the prodrome, to distinguish them from those characteristic of infection (e.g. paralysis for poliovirus, hemorrhagic fever for Ebolaviruses, rash for measles virus).
Whether or not an infected person is contagious (i.e. is shedding virus) during the incubation period depends on the virus. For example, Ebola virus infected patients do not pass the virus on to others during the incubation period. This fact explains why Tom Frieden said there was ‘zero chance’ that the passenger from Liberia who was diagnosed with Ebola virus infection in Dallas would have infected others while on an airplane. He had no symptoms of infection because he was still in the incubation period of the disease.
In contrast to Ebolaviruses, poliovirus and norovirus are shed during the incubation period – in the feces, where they can infect others.
Remarkably, viral incubation periods can vary from 1 or 2 days to years (Table; click to magnify). Short incubation times usually indicate that actions at the primary site of infection produce the characteristic symptoms of the disease. Longer incubation times indicate that the host response, or the tissue damage required to reveal the symptoms of infection, take place away from the primary site of infection.
The table was taken from the third edition of Principles of Virology. Missing from the table (which will be corrected in the next edition) is the incubation period of Ebola virus, which is 2 to 21 days. I would also argue that the incubation period of HIV is not 1-10 years, but 2-4 weeks, the time until the prodromal symptoms occur. The characteristic symptom of HIV-1 infection, immunosuppression, occurs much later.