In order to answer this question, we must first think about the meaning of the term “popular party.” I would say that a party is “popular” if it can win a lot of votes from people in free and fair elections. Therefore, in order to determine whether the Nazi Party was popular, we must look at election returns. This will be problematic because there were no free and fair elections in Germany after 1932.
There were seven federal elections in Germany from May of 1924 through March of 1933. Of these, all but the last can be seen as free and fair. The last election, in March 1933, was held while Hitler was already chancellor of Germany. Civil liberties had been suspended at that time because of the Reichstag Fire in February of that year. Communists had been arrested and the elections were not really free.
In these free elections, the Nazis’ fortunes rose and fell. In the first three elections, they never got more than 6.5% of the vote. This shows that they were not particularly popular through these elections, the last of which was in May 1928. After that, Germany fell into terrible economic trouble. This made the Nazis much more popular because they seemed to have an answer to the problems. In the next three elections, the Nazis got 18.3% of the vote (September 1930), 37.3% of the vote (July 1932) and 33.1% of the vote (November 1932). This means that, at their peak, the Nazis were favored by about one-third of the population. This makes them a party that was at least somewhat popular. When an election was held in less-than-free conditions in May of 1933, the Nazis got 43.9% of the vote.
After 1933, it is hard to know how popular the Nazis were since the country was now completely run by them and civil liberties were strongly curtailed. Evidence suggests that they were rather popular at least for a while as they seemed to be bringing Germany out of the Depression and making it stronger. The popularity of the party peaked as WWII started and Germany did well, but then declined as the war progressed.
Thus, the Nazis were never the choice of a majority of Germans in the time of free and fair elections. They were quite popular however, getting about one-third of the vote and having more seats in the Reichstag than any other party. Their popularity then rose (we think) while they were in power until WWII caused that popularity to fall.