a virus that, upon infection of a host cell, makes a DNA copy of its RNA and inserts that piece of DNA into the cell's genome
You are describing a type of virus called a retrovirus. Retroviruses are RNA viruses that have the unique ability to convert their RNA genome into DNA using the enzyme reverse transcriptase. This process is called reverse transcription.
Upon infecting a host cell, retroviruses use their own reverse transcriptase enzyme to convert their single-stranded viral RNA genome into a double-stranded DNA molecule. This newly formed DNA, known as cDNA (complementary DNA), is then integrated into the host cell's genome with the help of another viral enzyme called integrase.
Once integrated, the viral DNA becomes a permanent part of the host cell's genetic material. It is replicated along with the host DNA during cell division, ensuring that subsequent generations of daughter cells also contain the viral DNA.
The viral DNA in the host cell's genome can be transcribed and translated by the host cell's machinery, leading to the production of viral proteins and the assembly of new virus particles. These particles can then go on to infect neighboring cells and continue the viral replication cycle.
Examples of retroviruses include Human Immunodeficiency Virus (HIV), which causes AIDS, and Human T-cell Leukemia Virus (HTLV), which causes certain types of cancer.