To build on T.E.D's excellent answer, it is important to understand the backdrop under which Wallace was nominated for Vice-President in 1940, and not nominated in 1944.
The first thing to note, was that as late as the 1930s and 1940s, the Republican party was the "centrist" and "Establishment" (but pro-business party), while the Democrats were an unlikely mix of left AND right. That is, it included "everyone" who was not a Republican, whether urban laborer, urban liberal, progressive agrarian, or a fundamentally right-wing member of the agricultural "old South." (The Republicans represented the northeastern industrial and commercial elite, but not the agricultural elite from elsewhere).
When Franklin Delano Roosevelt (FDR) was first nominated in 1932, he was a "plurality" (not majority) "left" Democratic candidate from New York, who was put "over the top" by John Nance Garner, a conservative Texan. (Garner decided that he wanted "second prize" for sure.) As such, it was a perfectly balanced "left-right" ticket that could run on "competence, not ideology," against the failed but centrist Republican Administration of Herbert Hoover.
Garner didn't support FDR's more progressive initiatives in the later stages of the "New Deal," notably opposing FDR's attempt to "pack" the Supreme Court with left-leaning justices, which is why FDR replaced him as a VP candidate with Henry A. Wallace, who was even further left than FDR, in 1940. Only FDR's two previous successful terms (and threat not to run for a third one) caused the Democrats to accede to this.
Wallace turned out to be exactly the ultra-leftist that most Democrats feared. In 1943, he expressed sympathy with rioting African-Americans, which troubled most of the United States, but particularly the South. He also visited the Soviet Union in 1944, and expressed views that made other Administration members regard him as a Stalinist "stooge."
If he had been nominated as Vice-President (and a prospective President, given FDR's decline) in 1944, it would have split the Democratic Party into right and left wings, and (probably) handed the Presidency to the Republicans. Instead, the Democrats nominated the far more pragmatic Harry S. Truman, who came from Missouri, a "border" state that was almost equally acceptable to the Western and Southern parts of the country, shoring up FDR's electoral power in those two regions. Truman held them in his own election in 1948, losing FDR's northeastern base to Dewey.