It's hard to precisely count the number of political parties that existed, and there probably isn't a comprehensive list of every single minor faction. If we just use the number of parties that received votes in at least one election between 1919 (the first election under the Weimar constitution) and March 1933 (the last multi-party elections in a unified Germany until 1990), there were at least 160 parties that were active on a significant level during the Weimar period. Note that some of these parties were reformed versions of an existing party, splinter groups of a larger party, or mergers of two or more pre-existing parties, so these weren't necessarily 160 totally distinct political entities.

Of course, the majority of these parties were very minor. They were mainly special interest parties (trade groups, ethnic minority parties, etc.) which were never going to win a large number of seats or enter into government. There were a few major splinter groups that were of genuine electoral significance, such as the Independent Social Democratic Party (USPD), which became the second-largest party in the Reichstag (behind the "main" Social Democratic Party) following the 1920 election.

Of the 160 or so parties that participated in election(s) in Weimar Germany, only 29 won at least one seat in the Reichstag at some point between 1919 and March 1933, and there were 11 parties in the Reichstag after the March 1933 election. By the time of the November 1933 election, all political parties aside from the Nazi Party had been banned, and the only option was to vote for or against the Nazi list, which officially received over 92% of the vote.

There are 7 parties that are generally accepted as "major" parties (listed from left to right on the political spectrum): the Communist Party, the Social Democratic Party, the German Democratic Party, the Center Party, the German People's Party, the German National People's Party, and the Nazi Party. A few of the minor parties participated in some of the coalition governments during the Weimar era as well.

As far as comparisons to other political systems, the legislative electoral system in Weimar Germany was very different from both the first-past-the-post American system and the mixed-member-proportional system used in German federal elections today. Under the Franchise Law of 27 April 1920, Weimar Germany used a simple system of proportional representation, where citizens voted for a political party and members of the Reichstag were allocated from party lists from each of the 35 electoral districts. A party had to win at least 60,000 votes to win a seat in the Reichstag, which meant that the number of seats fluctuated from election to election based on turnout and the number of parties that crossed the threshold (from as few as 423 seats in 1919 to 647 in March 1933). Because of the relatively low threshold, parties could gain seats in the Reichstag with relatively trivial percentages of the overall vote (in several cases, less than 1%).

This system was kind of a double-edged sword. It had the benefit of allowing small parties to gain representation in the Reichstag, but that meant it was practically impossible for one party to gain an absolute majority of seats, allowing them to govern alone. In fact, no party even came particularly close to winning a majority: the closest in a fair election was the SPD, which got 38% of the vote in the 1919 election, and even the Nazis couldn't gain a majority in March 1933, only winning about 44% of the seats despite rampant voter intimidation. As a result, the "winning" party had to form a coalition of several parties to create a government. This was obviously highly unstable, because if one party chose to leave the coalition, that coalition could lose its majority and the government would collapse, requiring a new election. In fact, no Weimar government ever survived a full four-year term; the longest period between elections was three and a half years, from December 1924 to May 1928, although there were four separate governments during that period, all of which were center-rightcoalitions led by the Center Party (DZP).

The question of whether the disunity of the non-Nazi parties in Weimar Germany prevented them from forming an effective bulwark against the Nazis is controversial, but since that wasn't really your question, suffice to say there were a number of reasons that a "popular front" against Nazism was unlikely and politically impractical, despite the fact that the Nazis were never the majority party after a free election.

Sources:

Fuad Aleskerov, Manfred J. Holler, and Rita Kamalova, "Power Distribution in the Weimar Reichstag in 1919-1933," Annals of Operations Research 215 (2014): pp. 25-37. (Information on the political parties that entered the Reichstag and the electoral system)

Deutscher Bundestag, "Elections in the Weimar Republic" (2006). (Information about electoral results)

Richard J. Evans, The Coming of the Third Reich (Penguin, 2004), pp. 339-344. (Information on the conduct of the March 1933 election and its consequences)

Dieter Nohlen and Philip Stöver, Elections in Europe: A Data Handbook (Nomos, 2010), pp. 746-790. (Statistical information on the results of elections and the electoral system)