Professor Ivy is a Pokemon Professor from the Orange Islands. Brock developed a crush on this... voluptuous woman, and he decided to reside with her at the Orange Islands. Later on, Brock returned to Ash's team, and that whenever anyone asked him about what happened between him and the young Pokemon Professor, he would always curl into a ball, and he would desperately try to beg the person who asked to not mention her name again.
Because of this noodle incident, many fans have posited theories to try to explain Brock's falling out with Ivy. These are those theories.
One theory states that Ivy was simply just not interested in him. While being turned down by a girl is nothing new to Brock, it is the way that Ivy rejected him that made him cringe whenever her name was mentioned to him. It's also believed that Ivy was either already married, or that she was already dating someone, most likely Professor Oak.
There are also many perverted theories as to why Brock's relationship with Professor Ivy didn't work out. Many people believe that Brock had gotten Ivy pregnant, and that he then fled the scene so that he wouldn't have to take responsibility for his mistake. However, this theory seems to fall apart once you remember his responsibility regarding his position as a Gym Leader back in his home town. Because of this fact, it would seem unlikely that Brock would leave her for this reason. It's even believed that when Brock and Ivy were alone together, she had made a comment to him regarding his small penis. This ultimately damaged his self-esteem, and he never went back to the Orange Islands because of his "inadequacy."
There are also many questions regarding Professor Ivy's true gender. Many fans of the anime believe that Ivy might have actually been a man at some point in her life, and Brock immediately found out about it and fled the scene. There have been many non-canonical fan comics that parody this idea. For example, in several of the comics, a common theme would be that a Pokemon that she and her assistants were working on reversed their gender roles, and that Ivy and her assistants would then return to their true selves at the end of the comic.
According to one official source, the reason as to why Brock was initially retired in the first place was because many of the producers of the Pokemon anime assumed that people in western countries would interpret Brock as being a racial stereotype because of his squinted eyes. This explanation also explained why Tracey was made after Brock left, and it also explains why Brock came back later on from his stay there. However, there is still no official word as to why Brock left Ivy, nor will there ever be one.