How many bowl games are there?

Decades ago, an invitation to a bowl game was special because only a select few schools got to play in one. In 1966, there were only nine bowls in major college football. More than 50 years later, there are 41.

What are the requirements for a bowl?

In general, a college football team must have at least six wins. There have been some cases when 5-7 teams qualified for bowls, such as Nebraska in 2015.
Conference affiliation plays a large factor in bowl games, and Nebraska's Big Ten connection helped it sneak into postseason play that season.  

What is the history of bowls? 

In 1955, there were only seven bowls. In 1980, there were 15. In 1997, there were 20.

The reason there are more than twice the number of bowls today than in the late 1990s is because of a four-letter word: ESPN. According to a December 2017 Business Insider article:

Of the 41 games, 36 will be broadcast by ESPN or one of its sister networks, ABC and ESPN2. The reason ESPN is willing to pay to broadcast as many bowl games as possible is that live sports makes ESPN the king of cable, and the bowl games fill empty time slots. To up the ante, ESPN's subsidiary, ESPN Events, now owns 13 of the bowl games, according to USA Today, up from just three bowl games 12 years ago.