Any type of struggle, incompatibility, or discrepancy between the main character and something outside of the character's control is considered an "external conflict." Whenever the goals or objectives of the main character come in direct conflict with something the main character cannot fix or change, the conflict becomes "external."
That said, these types of conflicts could occur from anything that does not come from "within:" between the key character and another character, a force of nature, or as a result of a given situation.
In Flowers for Algernon, there are plenty of external conflicts, but they are all as a result of one central conflict: Charlie versus the social ignorance of special needs.
The overall ignorance of special needs is what ignites every conflict in the novel. First, we see it in the medical community's inability to understand what is at stake for someone like Charlie to undergo a change such as the one that he made.
While the operation in the novel is fictional, consider how unethical it would be to conduct such an operation even if it were available. There will never be a true consent from a patient to allow for it to occur because the patient would lack some of his or her mental capacities to make such a complex choice. The rights of the patient and uncertainty of the outcome would make it all the more dangerous and questionable.
We also see ignorance in the way Charlie continues to be a social misfit whether he is intellectually low or high. This is a clear sign the society depicted in the novel is unwilling or unable to appreciate (or tolerate) different people, or those with different needs that deviate from what is considered "the norm."
The needs of cognitively different people being overlooked and misunderstood is the driving factor that moves the plot forward until its sad ending. This is, therefore, the biggest of all external conflicts in the novel.
There are several external conflicts in Flowers for Algernon.
One conflict involves Charlie and the scientists. Once Charlie becomes extremely intelligent, he wants to work to make sure that the process doesn’t reverse itself. The scientists are unable to do this, which causes tension between them.
There is also a significant conflict between Charlie and his love interest, Alice. As their relationship changes due to Charlie’s increased mental capacity, Charlie is not always able to observe boundaries. His emotional maturation is not quite in line with his mental development and when he tries to move too fast it strains their relationship.