There are currently 5 notable variants of the coronavirus causing the pandemic. The virus itself is identified as SARS-CoV-2 and is similar to the SARS-CoV-1 that had an outbreak during 2002-2004 causing SARS (the disease). SARS meaning severe acute respiratory syndrome and CoV meaning simply corona virus. The number just denoting the identification as major strain.
In addition to variants, which there are an additional 8 to those notable ones there are also different lineages, clades and mutations classifying the different versions of the virus. So while the base virus is the same there are countless different versions of it out there. Each one acting a bit differently.
COVID-19 is the actual disease caused by the virus (any version of it). The name comes unimaginatively from COrona VIrus Disease 2019.
The variants are named after Greek alphabets up until Mu after when the Xi and Nu were skipped. The reasoning ironically was similar to why these alphabets were originally chosen: to make them not unintentionally reference any group of people or locations. In the past similar naming has had some stigmata e.g. the spanish flu, China flu. Even the first variant of SARS-CoV-2 was initially referred as Wuhan-1 after the place it was first discovered. To ensure any possible new viruses will have a non-stigmata names the WHO has decided to use constellation names in the future.