Natural gas is just really, really old biogas
The terms "natural gas" and "biogas" both describe a mix of gaseous hydrocarbons primarily composed of methane; the different names depend on the origin. According to the U.S. Energy Information Administration:
Natural gas occurs deep beneath the earth's surface. Natural gas consists mainly of methane, a compound with one carbon atom and four hydrogen atoms. Natural gas also contains small amounts of hydrocarbon gas liquids and nonhydrocarbon gases.
In contrast (from Wikipedia):
Biogas typically refers to a mixture of different gases produced by the breakdown of organic matter in the absence of oxygen. [...] Biogas is primarily methane (CH4) and carbon dioxide (CO2) and may have small amounts of hydrogen sulfide (H2S), moisture and siloxanes.
Propane is a by-product of natural gas extraction
Unlike natural gas and biogas which are mixtures of methane plus various other gases, propane is just one gas (from Wikipedia):
Propane is a three-carbon alkane with the molecular formula C3H8. [...] A by-product of natural gas processing and petroleum refining, it is commonly used as a fuel.
Natural gas, biogas, and methane are interchangeable*, propane is not
Because natural gas and biogas are primarily composed of methane, you can generally use them interchangeably.
*That said, certain types of equipment (whether combustion generators or otherwise) likely require a certain minimum methane concentration, which will vary for natural gas and biogas depending on the source.
However, propane is an entirely different chemical compound, meaning equipment designed for propane will most likely not work with any other gas.
A note on fuel cells
From the question:
It seems that there are some projects that are using generators to convert the methane gas into electricity, however, I can't tell if this is special kind of generator and I just don't know what it's called or if it is a natural gas or propane generator.
It sounds like you may be referring to fuel cells. From Wikipedia:
A fuel cell is an electrochemical cell that converts the chemical energy from a fuel into electricity through an electrochemical reaction of hydrogen fuel with oxygen or another oxidising agent.
There are a number of different fuel cell technologies, some of which involve extracting hydrogen from methane, propane, or other hydrocarbons.