Viruses are unique microbes in that they aren't really alive. They exist in two forms: a form outside the cell called the virion and a form inside the cell called a virus. A virus is the replicating form and only exists inside a cell. The virus hijacks the host cell machinary to make more virions. The virions are released from the cell to infect the next cell.
All viruses consists of protein and nucleic acid. Protein makes up the structure of the virus as well as aiding host cell hijacking. The nucleic acid is the genetic information of the cell. Like all cells, we humans have DNA as our genetic material. Since viruses are not cells, they are not bound by the DNA = genetic material law. Viral genomes can be made of RNA or DNA. The DNA or RNA can have a different structure than cells as well. In cells, DNA is double-stranded and RNA is single stranded. For viruses, the DNA or RNA can be double-stranded or single-stranded. Virologists use the nature of the nucleic acid to classify viruses.
A virus will use the protein component to enter host cells and hijack the host cell machinary using the nucleic acid as the directions to make more virions. The replication of the virus often has negative consequences on the host cell. The location of the host cell and the number of infected cells in turn has an affect on the host. The affect on the host is what produces the symptoms and consequences of a viral infectious disease. I'll illustrate this process with each virus we discuss.
Section 2 of Medical Microbiology describes virus structure, replication and pathogenesis in more detail.